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Modeling and dynamic analysis of coupled structure-moving subsystem problem
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Modeling and dynamic analysis of coupled structure-moving subsystem problem
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Content
MODELING AND DYNAMIC ANALYSIS OF
COUPLED STRUCTURE-MOVING SUBSYSTEM
PROBLEM
By
Hao Gao
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Fulfillment of the Requirement for the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)
August 2019
i
Acknowledgements
I would like to express my deepest thanks to my academic advisor, Dr. Bingen Yang, for his
guidance and encouragement throughout my graduate studies. I am also grateful to the members
of my Ph. D. dissertation committee: Dr. Henryk Flashner and Dr. Carter Wellford for their
constructive suggestions and advice. Meanwhile, I want to thank my lab member, Ruiyang Wang,
for his active idea exchange.
At last, I want to give my greatest thanks to my dear parents, who support me all the way
through my study at USC. I would not be able to accomplish my work without their support and
encouragement. Mom and dad, I love you forever.
ii
Table of Contents
Acknowledgement………………………………………………………………………………...i
Table of Contents………………………………………………………………………………...ii
Abstract…………………………………………………………………………………………..vi
Chapter 1 Introduction ............................................................................................................. 1
1.1 Motivation and Purpose .................................................................................................. 1
1.2 Literature and Background ............................................................................................. 2
1.3 Summary ......................................................................................................................... 7
Chapter 2 Beam-Moving Oscillator Problem ......................................................................... 9
2.1 Introduction ..................................................................................................................... 9
2.2 Problem Statement ........................................................................................................ 11
2.2.1 Four stages of beam-oscillator interactions ............................................................ 12
2.2.2 Governing equations ............................................................................................... 16
2.3 Semi-Analytical Modeling and Solution Method ......................................................... 18
2.3.1 Analytical eigensolutions ........................................................................................ 19
2.3.2 Extended solution domain....................................................................................... 22
2.3.3 Dynamic response by generalized assumed-mode method .................................... 24
2.4 Numerical Examples ..................................................................................................... 27
2.4.1 Transient response .................................................................................................. 29
2.4.2 Convergence study .................................................................................................. 36
iii
2.4.3 Comparison with FEM ............................................................................................ 40
2.5 Conclusion .................................................................................................................... 43
Chapter 3 Beam-Moving Rigid Body Model ........................................................................ 46
3.1 Introduction ................................................................................................................... 46
3.2 Problem Statement ........................................................................................................ 47
3.2.1 Description of the passage of rigid bodies .............................................................. 49
3.2.2 Governing equations of motion .............................................................................. 51
3.3 Semi-Analytical Modeling and Solution Method ......................................................... 54
3.3.1 Analytical eigensolution for multi-span beam ........................................................ 54
3.3.2 Extended solution domain....................................................................................... 57
3.3.3 Approximate model by generalized assumed-mode method .................................. 60
3.3.4 Dynamic response of the coupled system ............................................................... 63
3.4 Extension of the Proposed Method ............................................................................... 64
3.4.1 Rigid bodies with time-varying speeds ................................................................... 65
3.4.2 Moving rigid bodies with multiple suspensions ..................................................... 66
3.4.3 Beam structure partially supported by viscoelastic foundation .............................. 69
3.5 Numerical Examples ..................................................................................................... 70
3.5.1 A simply-supported beam carrying one vehicle ..................................................... 70
3.5.2 Effects of vehicle speed and size on the vibration of a four-span beam ................. 72
iv
3.5.3 Three vehicles moving over a multi-span beam partially supported by viscoelastic
foundation ............................................................................................................................ 76
3.5.4 Parametric resonance by a sequence of moving rigid bodies ................................. 80
3.6 Conclusion .................................................................................................................... 87
Chapter 4 Parametric Resonance Analysis........................................................................... 90
4.1 Introduction ................................................................................................................... 90
4.2 Problem Statement ........................................................................................................ 92
4.2.1 Governing equations ............................................................................................... 93
4.2.2 Description of periodic passage .............................................................................. 94
4.3 Modeling and Solution Methods ................................................................................... 95
4.3.1 Generalized assumed-mode method formulation ................................................... 96
4.3.2 Mapping transformation formulation ...................................................................... 98
4.3.3 Highlights of the mapping transformation method ............................................... 102
4.4 Numerical Results ....................................................................................................... 102
4.4.1 A benchmark problem for beam-moving oscillator system .................................. 102
4.4.2 Effect of number of modes on parametric resonance criterion ............................. 108
4.4.3 Transient response of induced structure vibration ................................................ 111
4.4.4 Effect of modal damping on parametric resonance .............................................. 116
4.5 Discussion and Conclusion ......................................................................................... 119
4.5.1 Extension of the proposed method ........................................................................ 119
v
4.5.2 Conclusion ............................................................................................................ 121
Chapter 5 Dynamic Analysis of a Fast Projection System ................................................ 123
5.1 Introduction ................................................................................................................. 123
5.2 Problem Statement ...................................................................................................... 125
5.3 Solution Method.......................................................................................................... 128
5.3.1 Extended solution domain..................................................................................... 129
5.3.2 Analytical eigensolutions of a cantilever beam .................................................... 131
5.3.3 Generalized assumed-mode method ..................................................................... 133
5.4 Numerical Results ....................................................................................................... 137
5.4.1 Effect of acceleration and launching rate .............................................................. 137
5.4.2 Parametric resonance projectile induced vibration ............................................... 141
5.5 Summary ..................................................................................................................... 145
Chapter 6 Conclusion ........................................................................................................... 146
Reference…………………………………………………………………………………...….150
vi
Abstract
A semi-analytical method, based on distributed transfer function method (DTFM) and
generalized assumed-mode method, is developed for modeling and dynamic analysis of coupled
structure-moving subsystem problem. This newly developed method has a variety of engineering
applications including elevated railways and highway ramps, cable transportation system, fast tube
transportation and weaponry systems. Also devised in this effort is an analytical method of
mapping transformation that can predict the parametric resonance of a structure induced by
repeatedly passing subsystems.
The system in consideration is a coupled distributed-lumped system. The distributed system is
a flexible supporting structure, which in this work is modeled as a stepped multi-span beam with
flexible column supports. The lumped systems are moving subsystems, such as oscillators or rigid
bodies, which pass over the supporting structure. The distributed and lumped systems are coupled
at their interconnecting points by springs and dampers. The coupled system is governed by a mix
set of one partial differential equation and several ordinary differential equations. Due to time
varying locations of moving subsystems, the coupled system is nonautonomous or time-variant
whose general analytical solution is not accessible. What’s more, flexible coupling between the
structure and moving subsystems is extremely complicated because the number of contact points
for one subsystem, and total number of subsystems in contact with the structure are generally time
varying. To have a consistent and systematical formulation for structure-vehicle interaction, an
extended solution domain (ESD) technique is developed. The ESD is a union of three domains,
the beam domain and two virtual domains. The structure is extended to the virtual domains as rigid
vii
surfaces. Within the ESD, all moving subsystems are coupled with the “extended structure”,
rendering a fixed number of DOFs and constant matrix dimensions in formulation.
To solve the dynamic response of the coupled system, a semi-analytical method, that constituted
of DTFM and generalized assumed-mode method, is developed. Analytical and closed form
eigenfunctions (mode shapes) of the multi-span beam is obtained with augmented DTFM
formulation. The coupled system is discretized via application of generalized assumed-mode
method with analytical mode shapes being comparison functions. Because of this, the proposed
method is proved to be much more efficient than conventional FEM. Numerical results reveal that
vibration of the structure is dominated by static deflection caused by equivalent force if subsystems
are moving relatively slowly. Dynamic interaction between the structure and moving subsystems
will be significantly increased if subsystems are moving fast. Vibration of the structure with ever-
increasing and significantly large amplitude can be induced by repeated passage of moving
subsystems, which is identified as parametric resonance.
Parametric resonance is different from conventional resonance where the frequency of external
excitation matches with one of system’s natural frequencies. It is caused by repeated changing of
system configuration, which is the repeated passage and coupling of moving subsystems for the
problem studied in this work. A semi-analytical prediction method is developed based on mapping
transformation. Quantitative resonance criterion is established by evaluating the spectral radius of
a mapping matrix derived for the coupled system. Evaluating of the mapping matrix only requires
simulation for one period of passage. This makes the proposed method extremely efficient in
determination of parametric resonance. The proposed method is capable to predict not only the
resonance condition, but also the steady state value for a bounded response. Numerical results
show that the parametric resonance induced by a sequence of moving subsystems highly depends
viii
on system parameters, such as speed and spacing distance of subsystems, and cannot be accurately
predicted by using a one-mode approximation for the beam structure.
1
Chapter 1 Introduction
1.1 Motivation and Purpose
Flexible structures supporting multiple moving subsystems are commonly seen in many
engineering applications. Examples are diverse, including highway bridges, elevated guideways,
and railways with moving vehicles [1,2], cable transport systems [3,4], automobile disk brakes [5],
high-speed trains on viaducts, and tubes conveying fast-moving pods. When multiple subsystems
move over a flexible structure, dynamic interactions between the structure and subsystems take
place, which can generate significant large deflection of and stress in the structure. Due to the
dynamic interactions, response of the coupled system may be very different from that of a
corresponding structure subject to equivalent self-weight. In this regard, accurate modeling and
vibration analysis are therefore essentially important to reliable design and safe operation of this
kind of coupled flexible structure-moving bodies system.
In studying the dynamic response of a structure-moving bodies system, accurately modeling
the interaction between the structure and moving subsystems is the key issue. Inertia effect of a
subsystem and elastic coupling between the subsystem and the structure need to be considered.
Meanwhile, a coupled mathematical model, which combines a distributed system (i.e., the
structure) and many lumped systems (i.e., the subsystems), must be derived and solved
simultaneously. What’s more, the coupling due to moving subsystems renders possible
mathematical models time-variant, which has no analytical solutions generally. All those issues
make formulation, analysis and solution of the coupled structure-moving subsystems problem
extremely difficulty by a conventional method, if not impossible.
Additionally, subsystems passing over the structure sequentially could behave like a periodic
excitation and induce significantly large vibration of the structure. Unlike conventional resonance
2
that occurs when the frequency of external excitation is equal to one of the natural frequencies of
the system, resonant vibration of the structure induced by moving subsystems is highly dependent
on system parameters and passage pattern of moving subsystems because flexible interaction
between the structure and moving subsystems are considered. It requires an effective method to
predict the parametric resonance caused by moving subsystems and provide corresponding
parametric analysis.
The purpose of this study is to develop a method to model and solve the coupled structure-
moving subsystems problem systematically. Several types of subsystem-structure interaction
model are discussed and validated with numerical simulations. Besides general derivations and
solutions, a novel semi-analytical method based on mapping transformation is developed to predict
the parametric resonance caused by moving subsystems. The goals of this study are summarized
as followings:
(1) To derive accurate mathematical models for the coupled structure-moving subsystems
problem;
(2) To develop a semi-analytical solution method which solves the dynamic response for the
coupled system;
(3) To develop a semi-analytical mapping method to predict the parametric resonance induced
by moving subsystems.
1.2 Literature and Background
Modeling and vibration analysis of a structure carrying moving subsystems has been
extensively studied in the past. In the literature, three basic types of problems have been addressed:
(a) moving load problems, (b) moving mass problems, and (c) moving oscillator problems. In a
moving load problem, a subsystem is modeled as a concentrated force due to gravity that moves
3
along the supporting structure and the inertia effect of the subsystem is neglected. In a moving
mass problem, the inertia of the subsystem is included, but the connection between the subsystem
and the structure is assumed to be rigid. As such the subsystem practically slides along the structure
while undergoing transverse vibration. In a moving oscillator problem, the subsystem is described
as a spring-mass-damper system that is elastically connected to the supporting structure during its
motion. A dynamic system in a moving oscillator problem is time-varying, with its inertia, stiffness
and damping parameters being time-dependent. In most cases, moving oscillator problems provide
results that are physically closer to the reality than moving load and moving mass problems.
For moving load problems, various beam models and solution methods have been proposed.
Early works on the dynamic response of a finite beam subject to moving loads adopted both Euler-
Bernoulli and Timoshenko beam models [6,7], and complex Fourier transformation method was
shown to be able to give closed-form solutions of beam response. Investigation of moving load
problems was extended to beams of infinite length, and two- and three- dimensional solids, as
demonstrated by Fryba [8]. The dynamics of a multi-span beam under moving loads was studied
by Henchi and Fafard through use of exact stiffness element method [9] and by Zheng et al. [10]
with a modified cubic polynomial eigenfunction method. Song et al. [11] applied a frequency-
domain spectral element method to study the dynamics of a Timoshenko beam under moving loads
of time-varying speeds. Jorge et al. studied the dynamics of beams on non-uniform nonlinear
foundations subjected to moving loads with FEM formulation [12]. Some preliminary work of
resonant response induced by moving forces has also been done in recent years. Museros et al. [13]
studied the resonance of a simply-supported beam bridge under moving forces and its cancellation
condition. Yang and Yau [14] investigated the resonance response of a series of beams bridges
carrying moving vehicles by simplifying a vehicle as a concentrated force.
4
Compared with the moving force model, moving mass model considers the inertia effect of a
subsystem, allowing to analyze the vibration of the subsystem itself which is important for
passenger comfortability. For moving mass problems, several modeling and analysis methods have
been developed, including influence function method [15,16], Lagrange multiplier method [17],
assumed-mode method [18], and modal expansion method [19-21]. Mofid and Akin [22] proposed
a discrete element method to determine the response of a Euler-Bernoulli beam with traveling mass,
which was extended to Timoshenko beams by Yavari [23] et al. Amiri et al. [24] also applied the
modal expansion method to a plate structure subject to moving mass. Galerkin method was used
by Karimi and Ziaei-Rad [25] to analyze the vibration of a beam with moving support subjected
to moving mass. Dimitrovova developed a semi-analytical solution for a beam on a two-parameter
viscoelastic foundation subject to a moving mass with assumed-mode method [26]. In the previous
investigations, the modal expansion method has been shown to be efficient in dealing with simply-
supported single-span beams and plates under moving mass.
However, a moving force or a moving mass model fails to describe the flexible interaction
between the structure and a subsystem because the inertia effect of a subsystem is neglected and
rigid contacting between the structure and subsystem is assumed. It becomes necessary to consider
moving oscillator or moving rigid body model, in which the inertia effect of a subsystem and
elastic coupling between a structure and subsystems are included. Since the 1990’s, different
modeling techniques and solution methods have been developed. Klasztorny and Langer [27] used
Lagrange-Ritz energy method to formulate a discrete equilibrium equation for a coupled bridge-
vehicle system, which is modeled as a simply supported Euler-Bernoulli beam carrying moving
oscillators. With a finite element formulation, Lin and Trethewey [28] found that higher order
modes of a supporting beam can be excited and rotary inertia should be considered in the analysis.
5
Yang et al. [29,30] developed a vehicle-bridge interaction element with FEM formulation and used
a condensation method to improve simulation efficiency. Pesterev and Bergman [31,32] proposed
a kernel integral equation method for beams under oscillator moving with constant speed and
acceleration. Yang et al. [33] and Bergman et al. [34] developed a numerical procedure to study
the dynamic response of a single-span one-dimensional continuum carrying moving oscillators.
Green and Cebon [35] investigated a simply supported bridge traversed by a vehicle modeled as a
single degree of freedom oscillator and established a parametric study with iterative simulation.
Rajabi et al. [36] studied the dynamic response of a functionally graded simply-supported Euler-
Bernoulli beam subject to a moving oscillator. They reduced the coupled system to a system of
second-order ordinary differential equations with the Galerkin method and solved results using
Runge-Kutta scheme. With a combined finite element and analytical formulation, Zrnic [37]
investigated transverse and longitudinal vibration of a gantry crane system subject to the elastically
suspended moving body. Wu and Gao [38] studied a viscous damped double-beam system under
a moving oscillator by applying modal expansion formulation and investigated the effect of some
system variables. Most recently, Yang and Gao [39] developed a semi-analytical method to solve
for dynamic response of a multi-span beam structure supported by columns subject multiple
moving oscillators, making it possible to analyze the transient response of a structure induced by
a sequence of vehicles passing by. Besides the above-mentioned studies, other important issues in
moving oscillator problems have been addressed, including single-span beams with flexible
boundaries [40], moving oscillator with random parameters [41], and the effects of a moving
oscillator during separation from and reattachment to the supporting beam [42].
In those studies, a moving subsystem is modeled as a single degree of freedom oscillator which
is a conventional quarter model for vehicles. Although the investigations on the moving oscillator
6
problem have provided some physical insights into vibrations of coupled structure-vehicle systems,
the dimensions or sizes of vehicles are ignored, and only one degree of freedom is adopted in
vehicle models. To accurately study the dynamic response of a moving vehicle and its effect on
the structure, consideration of vehicle sizes and multi-DOF vehicle model is necessary. In recent
years, rigid body vehicles models have been utilized by researchers. By using modal superposition,
Law and Zhu [43] studied the effect of road roughness and braking on a simply supported beam
under one moving rigid body. With finite element formulation, Lou [44] studied the dynamic
response of train-track-bridge interaction system under a single rigid body and force distribution
of it induced by multiple rigid bodies [45]. Through comparison of a 3-DOF rigid body vehicle
model with a moving load model, Liu et al. determined the critical speed of a vehicle at which the
maximum beam vibration occurs [46]. Chen at al. studied vertical vehicle-track interaction with
condensation method considering rigid body vehicle with four wheelsets [47]. With Lagrange
multiplier method, Fedorova and Sivaselvan studied the effect of train speed on bridge response,
as well as the contact separation between the bridge and train [48]. Zeng et al. [49] conducted a
comparison of a sprung mass model and a multibody model of trains, in which the supporting
structure is a three-dimensional bridge modeled by the finite element method. By establishing a
train-bridge model with consideration of full wheel and rail profiles, Olmos and Astiz studied the
critical train and wind velocities, at which the train cannot travel safely over the O’Eixo Bridge
[50]. Wang et al. proposed an iterative method to determine the dynamic response of a coupled
railway vehicle-track system [51], for which a track-vehicle interaction model based on three-
dimensional geometry relation was created.
Instead of the rich literature on structure-moving vehicle systems, deep understanding of the
dynamic behaviors of the coupled system still requires continued research. For a coupled system,
7
the interactions between a supporting structure and moving subsystems can be very complicated.
This is because the number of contact points between moving subsystems and supporting structure
is time-varying and generally arbitrary, which makes modeling and analysis of a coupled system
an extremely difficult task. This task becomes even harder if partial contact between a subsystem
and the supporting structure is detailed when the subsystem enters or leaves the structure, or if a
multi-DOF and multi-contact-point vehicle model is used. Moreover, parametric resonant
vibration of a supporting structure can be induced by a sequence of many moving subsystems.
Without an efficient technique to deal with complex structure-vehicle interactions, this type of
physical phenomena cannot be well investigated.
1.3 Summary
In chapter 2, a beam-moving oscillator model is demonstrated. For this case, the supporting
structure is modeled as a stepped Euler-Bernoulli beam with column supports, and each moving
subsystem is modeled as 1-DOF spring-mass-damper oscillator. Through extended Hamilton’s
principal, governing equations for the coupled system is derived using energy functionals, which
is a combination of a partial differential equation and a set of ordinary differential equations. An
extended solution domain (ESD) technique is developed to handle the coupling issue for an
arbitrary number of moving oscillators. With the ESD and generalized assumed-mode method, a
systematic formulation of discrete governing equation is derived whose solution delivers the
dynamic response for the coupled structure-moving oscillator problem.
In chapter 3, a 2-DOF rigid body model is applied for the moving subsystem, with which both
translational motion and rotational motion of the subsystem are considered. The coupling between
rigid bodies and the supporting structure is more complicated than that for moving oscillators
because multiple contact points exist for rigid body model and partial coupling could happen when
8
a rigid body is entering or leaving the supporting structure. With the ESD technique developed, a
consistent formulation for the coupling between moving subsystems and supporting structure is
derived in a systematic manner which allows easy implementation and numerical simulation. In
this part, some preliminary results of parametric resonance are presented.
In chapter 4, a novel semi-analytical method based on mapping transformation is developed to
predict the parametric resonance induced by moving subsystems. Based on the coupled beam-
moving oscillator model, the mapping transformation is formulated through the generalized
assumed-mode method in terms of state equations. The stability criterion or resonance criterion is
determined by the spectral radius of the mapping matrix. With the proposed method, the parametric
resonance induced by moving oscillators can be easily predicted by evaluating the mapping matrix
which depends on the system parameters.
In chapter 5, an extended case for the coupled structure-moving subsystem problem is presented.
In this case, the structure is a cantilever beam which describes shooting or projection systems. The
projectile or bullet is modeled as a 2-DOF rigid body. An augmented extended solution domain
(ESD) is defined for the projection system to deal with the coupling between the projectiles and
the supporting structure. Conclusions are addressed in chapter 6.
9
Chapter 2 Beam-Moving Oscillator Problem
2.1 Introduction
This chapter is concerned with the dynamic analysis of a coupled structure-moving subsystem
problem with a beam-moving oscillator model. As discussed in the previous chapter, three basic
models have been developed in literature: (a) moving load; (b) moving mass and (c) moving
oscillator. In early research efforts, researchers mainly focused on moving load problems [6-8]
during the 60s and 70s, and moving mass problems [15-20] from the 70s to 90s. Without
considering the inertia effect of a vehicle or elastic coupling between a subsystem and the structure,
moving load and moving mass models cannot accurately describe the dynamic interaction between
a supporting structure and moving subsystems. Under the circumstance, a moving oscillator model
which simplifies a vehicle as a 1-DOF spring-mass-damper oscillator becomes necessary. Since
the 1990s, vast literature [28-38] have been addressing the dynamic analysis of structure-moving
vehicle system with the oscillator model.
Even moving oscillator problems have been studied for decades, they mainly concentrated on
the case where one or just a few oscillators are considered. However, the dynamic response of a
structure induced by a sequence of many oscillators passing over might become significantly
different than that induced by just a few. Because of multiple oscillators moving at different speeds
and with varying spacing distances, the number of oscillators traveling on the supporting beam
structure is time-varying, which renders the number of degrees of freedom of a mathematical
model for the coupled system changing with time. It makes formulation, analysis, and solution of
the moving oscillator problem extremely difficult by a conventional method, if not impossible.
Furthermore, the dynamic response of a supporting structure carrying many moving oscillators
may be significantly different from that of a beam structure carrying only a few oscillators. Due to
10
repeated passage of oscillators, the vibration amplitude of the structure can become very large and
ever-increasing under certain conditions. Therefore, a systematic modeling and solution method is
important for the dynamic analysis of a structure carrying a sequence of moving oscillators. To the
author’s best knowledge, no such kind of techniques has been proposed in the literature.
In this chapter, a coupled beam-moving oscillator model is proposed for the structure-moving
subsystems problem. Unlike previous studies, the proposed model is capable to handle an arbitrary
number of oscillators systematically with a technique called extended solution domain (ESD)
method. The coupled distributed-lumped system is discretized by a generalized assumed-mode
method where the mode shape functions are analytical eigenfunctions for the beam obtained with
distributed transfer function method (DTFM) [52]. The discrete model is governed by a set of
second-order ordinary differential equations and can be further reduced to a set of state equations
by defining state vector in terms of generalized coordinates. Solving the state equations by
numerical integration gives the dynamic response of the coupled beam-oscillator system. As shall
be shown in several numerical examples, the proposed method can model a multi-span beam
structure coupled with arbitrarily many moving oscillators and is highly efficient and accurate in
computation.
The remainder of this chapter is arranged as follows. The moving oscillator problem and
relevant beam-oscillator interactions are described in Section 2.2. The proposed modeling and
solution methods are presented in Section 2.3. In Section 2.4, the proposed method is illustrated
on a four-span beam structure with a set of three oscillators and a platoon of 10 oscillators,
respectively, where numerical results validate the accuracy and efficiency of the method. Also, in
Section 2.4, the proposed method is shown to be able to save tremendous computation time,
11
compared to the finite element method. Finally, the main results and contributions from this
investigation are summarized in Section 2.5.
2.2 Problem Statement
A coupled beam-oscillator system in consideration is shown in Figure 2.1, where a multi-span
beam structure carries n moving oscillators. The structure of total length L is modeled as a stepped
Euler-Bernoulli beam with p elastic supports at the interior nodes with coordinates denoted by
,1 ,2 ,
, ,...,
s s s p
x x x , and thus it has (p + 1) uniform beam segments or spans. For the convenience of
analysis and solution, the origin of the coordinates is set at the left end of the beam. Besides the
supports at the interior nodes, the structure is also supported at its two ends (x = 0 and L), where
clamped or hinged boundary conditions are usually specified. Each of the oscillators is a spring-
mass-damper subsystem that moves at a constant speed, say
i
v . The dynamic interactions between
the beam structure and moving oscillators shall be described in Section 2.2.1.
Figure 2.1 Schematic of a multi-span beam structure carrying multiple moving oscillators.
The coupled system has three basic types of components as shown in Figure 2.2: (a) beam
segments, with density
l
, cross-section area
l
A , Young’s modulus
l
E , area moment of inertia
l
I , and span length
l
L ; (b) elastic column supports with height
j
h , Young’s modulus
, s j
E , cross-
section area
, s j
A and area moment of inertia
, s j
I ; and (c) moving oscillators, with mass
i
m ,
12
stiffness coefficient
i
k , damping coefficient
i
c , speed
i
v , and vertical displacement ( )
i
y t . The
governing equations of the components shall be devised in Section 2.2.2.
Figure 2.2 Basic components of the coupled beam-oscillator system:
(a) beam segment; (b) elastic column supports; (c) oscillator.
2.2.1 Four stages of beam-oscillator interactions
As oscillators move over the beam structure, dynamic interactions between the structure and
oscillators take place. There are four stages of the beam-oscillator interactions, which are portraited
in Figure 2.3 and explained in the sequel.
Stage I: Initial stage (t = 0)
Assume that n oscillators are to move over a multi-span beam structure from the left to right;
see Figure 2.3(a), where the oscillators are labeled from right to left, with the far-right oscillator
being the first one. Here integer n can be arbitrarily large. At the initial time t = 0, the oscillators
are all located on the left-hand side of the beam structure, with the first oscillator at the left end of
the beam structure (x = 0). The initial location of the ith oscillator is given by
i
x , with
1
0
1
1
0, , 2,3,...,
i
i l
l
d i n
(2.1)
13
where
0
l
d is the initial distance between the lth and (l+1)th oscillators, and x is measured from the
left end of the beam structure. The distance between the first and last oscillators then is calculated
as
1
0
1
n
L l
l
D d
. Thus, the last (far-left) oscillator is initially at
L
x D .
(a) Stage I
(b) Stage II
(c) Stage III-Case 1
(d) Stage III-Case 2
14
(e) Stage IV
Figure 2.3 Four stages of beam-oscillator interactions.
Stage II: Oscillators entering and staying in the structure domain (
1
0 min( / , / )
L n
t D v L v )
In this stage, the oscillators enter the beam structure one by one and no one leaves the structure;
see Figure 2.3(b). Assume that the speeds (
i
v ) and initial spacing distances (
0
l
d ) of the oscillators
are such that no two oscillators will occupy the same location at any time. In other words, no
collision between any two oscillators occurs. There are two parameters: /
L n
D v , which is the time
for the last (far-left) oscillator to arrive at the left end of the beam structure, with
L
D given in
Figure 2.3(a), and
1
/ L v , which is the time for the first oscillator to reach the right end of the
structure. The location of the ith oscillator at time t is given by
,
( )
o i i i
x t v t (2.2)
With the above “no collision” assumption, the distance between two adjacent oscillators is
always positive and it is given by
0
, , 1 1
( ) ( ) ( ) 0
i o i o i i i i
d x t x t v v t d
(2.3)
with 1, 2,..., i n and 0 ( ) /
L n
t D L v .
Stage III: Oscillators moving over the structure (
1 1
min( / , / ) max( / , / )
L n L n
D v L v t D v L v )
There are two cases in this stage, which are described as follows.
15
Case 1. All n oscillators have already been on the beam structure; see Figure 2.3(c), for which
the condition
1
/ /
L n
D v L v is satisfied. In this case, all the oscillators are continuously moving
until the first (far-right) oscillator reaches the right end (x = L) of the beam structure.
Case 2. Less than n oscillators are on the beam structure and the first oscillator has passed the
right end of the beam structure; see Figure 2.3(d), for which the condition
1
/ /
L n
D v L v is
satisfied. In this case, some oscillators are still entering the beam structure while others are leaving.
Stage IV: Oscillators exiting from the structure (
1
max( / , / ) ( ) /
L n L n
D v L v t D L v )
In this stage, all the n oscillators have entered the beam structure and some of them are exiting
from the structure; see Figure 2.3(e). This process continues until the last (far-left) oscillator
reaches the right end (x = L) of the beam structure.
To further understand the above-mentioned stages of the beam-oscillator interactions, view the
n oscillators as a platoon or worm that moves from left to right, with the head being the first (far-
right) oscillator and the tail being the last (far-left) one. In Stage I, the platoon head is at the left
end of the beam structure (x = 0); In Stage II, the platoon is entering the domain of the structure
with the head staying within the domain. In Stage III, the platoon is either on the structure entirely
(Case 1) or it is too long such that its head has already exited from the right end of the structure
before its tail arrives at the left end of the structure (Case 2). In Stage IV, the platoon continues to
move until its tail arrives at the right end (x = L) of the structure, at which time the platoon totally
moves off the structure.
From the previous discussion, the beam-oscillator interactions in these four stages are
complicated, with numerous possibilities of the number of oscillators on the beam structure at any
given time. This implies that the dimension of a mathematical model of the coupled system by a
conventional method may change frequently with time, which is especially true when the number
16
n of moving oscillators is arbitrarily large. Because of this complication in describing beam-
oscillator interactions, modeling and analysis of this type of coupled dynamic systems has been
limited to a supporting structure with just a few moving oscillators. As shall be seen in Section 2.3,
this issue can be completely resolved by a new method that uses an extended solution domain.
2.2.2 Governing equations
In this work, elastic column support is modeled as a pair of translational and torsional springs,
as shown see Figure 2.4. Here, without loss of generality, the inertia of the support is ignored. By
structural mechanics [53], the equivalent translational stiffness (
, t j
k ) and torsional stiffness (
, r j
k )
of the jth support are obtained as
, , , ,
, ,
,
s j s j s j s j
t j r j
j j
E A E I
k k
h h
(2.4)
Figure 2.4 Simplification pf elastic column supports.
Consider the four stages of the coupled system described in Section 2.2.1. By the Euler-
Bernoulli beam theory [54], the transverse displacement ( , ) w x t of the multi-span beam structure
supported by multiple translational and torsional springs is governed by
2 2 2
2 2 2
( , ) ( , )
( ) ( )
s o
w x t w x t
A x EI x f f
t x x
(2.5)
17
for 0 < x < L, where ( ) A x and ( ) EI x are linear density and bending stiffness of the multi-span
beam,
s
f is the resultant constraint force generated by the support springs and
o
f is the resultant
force due to interactions between the moving oscillators and the beam. The resultant constraint
and interaction forces can be written as
, ,
, , , ,
1 1
( , ) ( )
( , ) ( )
p p
s j s j
s t j s j s j r j
j j
w x t d x x
f k w x t x x k
x dx
(2.6)
,
, ,
1
( , )
( )
[ ( ) ( , )] ( ) ( )
n
o i
i
o i i o i i i o i
i
Dw x t
dy t
f k y t w x t c t x x
dt Dt
(2.7)
where
, ,
( ) ( ) ( )
i o i o i
t h x h x L is a function of time; ( ) and ( ) h are the Dirac delta function
and Heaviside step function, respectively; and ( )
i
y t is the vertical displacement of the ith
oscillator as shown in Figure 2.2(c). Displacement of the ith oscillator is governed by
2
,
, 2
( , )
( ) ( )
( ) [ ( ) ( , ) ( )]
o i
i i
i i i i i o i i i
Dw x t
d y t dy t
m c t k y t w x t t m g
dt dt Dt
(2.8)
The boundary conditions of the beam structure at x = 0 and L are of the form [55]
, ,
[ (0, )] 0, [ ( , )] 0, 1,2
L r R r
B w t B w L t r (2.9)
where
, ,
,
L r R r
B B are appropriate operators corresponding to certain boundary conditions. The
initial conditions for the coupled beam-oscillators system are described as follows
0 0
( ,0)
( ,0) ( ), ( )
w x
w x u x v x
t
(2.10)
( )
(0) , 0, 1, 2,...,
i i
i
i
m g dy t
y i n
k dt
(2.11)
where
0
( ) u x and
0
( ) v x are the initial displacement and velocity of the beam structure,
respectively. Here, without loss of generality, beam displacement is assumed to be measured from
its equilibrium position caused by self-weight, and each oscillator is assumed to be in its
18
equilibrium position under gravity and with zero vertical velocity. In other words, the oscillators
have no vibration before moving onto the beam structure. Of course, other initial conditions for
the moving oscillators can be prescribed without any difficulty and they shall not alter the proposed
solution process.
In this research, the fundamental problem is to determine the dynamic response of the coupled
beam-oscillator system, which requires solution of a mix set of the partial differential equation
(2.5) and ordinal differential equations (2.8), subject to the boundary conditions (2.9) and initial
conditions (2.10) and (2.11).
2.3 Semi-Analytical Modeling and Solution Method
There are four key issues in solving the dynamic problem as defined in the previous section.
First, the partial differential equations (2.5) and the ordinary differential equations (2.8) are
coupled, and they must be solved simultaneously. Second, the interactions between the beam
structure and moving oscillators result in a time-varying system, as implied by Eq. (2.7). Third,
the number n of moving oscillators can be arbitrarily large. Fourth, due to different speeds (
i
v ) of
moving oscillators and initial spacing distances (
0
l
d ), the number of oscillators on the beam
structure varies with time, as described by the four stages in Section 2.2.1. These issues make it
extremely difficult to apply conventional analytical or semi-analytical solution methods. Utility of
numerical methods, such as finite element method and finite difference method, besides having to
deal with many unknowns in a solution process, encounters the need to check the number of
oscillators on the beam structure at each calculation step, and to consequently change the solution
algorithm among many possibilities. Both analytical and numerical solution methods require
extensive and heavy computational efforts, especially when the number n of moving oscillators
becomes very large.
19
To address the above-mentioned issues, a new semi-analytical solution method is developed.
This method has the following three key components in the determination of the dynamic response
of the coupled beam-oscillator system:
(a) A distributed transfer function method (DTFM) is applied to obtain the exact eigenfunctions
of the multi-span beam structure (without oscillators).
(b) An extended solution domain is defined to treat the four interaction stages described in
Section 2.2.1 in a unified manner, which avoids frequent adjustment of solution algorithms due to
the time-varying number of oscillators on the beam structure.
(c) With the analytical eigenfunctions of the beam structure obtained in (a) and the extended
solution domain defined in (b), a set of state equations is established by a generalized assumed-
mode method. The state equations are then solved to give the dynamic response of the coupled
beam-oscillator system.
These key components of the proposed method are detailed in the sequel.
2.3.1 Analytical eigensolutions
In this effort, the eigensolutions of the n-span beam structure are obtained by DTFM [52,55,56].
To this end, instead of Eq. (2.5), consider the governing equation of motion of the lth beam
segment
2 4
2 4
( , ) ( , )
( , ), 0
l l
l l l l l l
w t w t
A E I f t L
t x
(2.12)
for 1,2,..., 1 l p , where ( , )
l
w t is the transverse displacement of the uniform beam segment,
, , ,
l l l l
A E I and
l
L are the beam parameters defined in Section 2.2, ( , )
l
f t is an external force,
and is the local spatial coordinate defined on the segment. The boundary conditions (2.9) are
redefined as
20
1 1 1
(0, ) ( , ) 0, 1, 2,3, 4
i i p p
M w t N w L t i
(2.13)
with
i
M and
i
N being appropriate operators.
Figure 2.5 Interconnection of beam segments and column supports at node j.
Because the beam structure is a continuum, two types of matching conditions are specified at
the interior nodes where two adjacent beam segments are interconnected: (i) displacement and
slope continuity, and (ii) force and moment balance. For the jth node, where the jth and (j+1)th
beam segments are interconnected and the jth column support is installed (see Figure 2.5), the
matching conditions are given by
1
1
, 1 1 1
, 1 1 1
( , ) (0, )
'( , ) '(0, )
'( , ) ''( , ) ''(0, )
( , ) '''( , ) '''(0, )
j j j
j j j
r j j j j j j j j j j
t j j j j j j j j j j
w L t w t
w L t w t
k w L t E I w L t E I w t
k w L t E I w L t E I w t
(2.14)
where ( )' ( ) / d d .
Recalling the DTFM formulation from Refs. 52, 55 and 56, taking Laplace transform of Eq.
(2.12) with respect to time t and casting the resulting equations into a spatial state form, yields
ˆ ˆ ˆ ( , ) ( ) ( , ) ( , ), [0, ]
l l l l l
s F s s p s L
(2.15)
where ˆ
l
is a state vector given by
2 3
2 3
ˆ ˆ ˆ ˆ ˆ ( , ) ( , ) ( , ) ( , ) ( , )
T
l l l l l
s w s w s w s w s
(2.16)
21
( )
l
F s is a four-by-four matrix consisting of the parameters of the jth beam segment; ˆ ( , )
l
p s
represents the external force and initial disturbances; s is the Laplace transform parameter, and the
hat (^) stands for Laplace transformation with respect to t. Also, the matching conditions (2.14)
can be converted into the s-domain state form
1
ˆ ˆ (0, ) ( , )
j j j j
s T L s
(2.17)
where the matrix
j
T consists of the parameters of the involved beam segments and column support.
The eigenvalue problem for the entire multi-span beam structure (without oscillators) can be
described by a homogeneous state equation that is obtained by vanishing ˆ ( , )
l
p s in Eq. (2.15)
and setting i s , with i 1 ; namely
( , ) ( ) ( , ), [0, ]
l
x F x x L
x
(2.18)
where is an eigenvalue (natural frequency) of the structure, and the state vector ( , ) x is the
associate eigenfunction. Then ( , ) x can be written as
( , ) ( , ) x x (2.19)
where is a constant vector to be determined, and ( , ) x is a state transition matrix given by
1
, 1 1 1 1
1 1
1 ,
1 1
,1
( )
1 , 1 ,
( )
,
1
(0, )
( , ) ( , )
( , )
j s j j j j
p s p p p p
F x
s
F x x T F L
F L
s j s j
F x x T F L
F L
s p
e
x x
x e e T e x x x
x x L
e e T e
⋯
⋯
(2.20)
It is easy to show that the state vector given in Eq. (2.19) automatically satisfies the matching
conditions (2.17). The state vector ( , ) x also needs to meet the boundary conditions (2.13) of
the whole beam structure. Thus, substituting Eq. (2.19) into Eq. (2.13) eventually yields the
following eigenequation
[ ( , )] 0 M N L (2.21)
22
where M and N are boundary condition matrices consisting of the coefficients in Eq. (2.13). (Refer
to Ref. 55 for the formulation of M and N.) It follows that the characteristic equation of the beam
structure is
det[ ( , )] 0 M N L (2.22)
Solving the transcendental equation (2.22) for yields the natural frequencies of the multi-
span beam structure. The eigenvector associated with a root is a non-trivial solution of Eq.
(2.21). With so determined, the mode shape ( ) x of the beam structure is given by
( ) [1 0 0 0] ( , ) x x (2.23)
In the above solution process for the eigensolutions, no approximation or discretization has
been made; the eigensolutions obtained by the DTFM are of exact and closed form.
2.3.2 Extended solution domain
As discussed in Section 2.2, moving oscillators with different speeds (
i
v ) and initial spacing
distances (
0
l
d ) render the number of oscillators traveling on the beam structure a time-varying
parameter. Because of this, there are numerous possibilities of oscillators entering and leaving the
beam structure. Thus, in a conventional solution process, an examination of the number of
oscillators on the beam structure and subsequent change of solution algorithms is necessary at each
computation step, which makes the determination of the dynamic response of the coupled system
complicated and time-consuming, especially when many oscillators are involved.
Figure 2.6 The extended solution domain of a coupled beam-oscillator system.
23
To deal with the issue of the time-varying number of oscillators on the beam structure, in this
work, an extended solution domain (ESD) is proposed. The idea behind the ESD is that a solution
process is undertaken in a domain with a fixed number of moving oscillators, with which number
checking on moving oscillators is not needed. To this end, define an extended solution domain Ω
as the union of three subdomains
B L R
where the subdomains are given as follows
Beam structure domain: { | 0 }
Left extended domain: { | 0}
Right extended domain: { | }
B
L L
R R
x x L
x D x
x L x L D
(2.24)
with
L
D given in Eq. (2.2) and
1
( )
R L
n
v
D L D L
v
(2.25)
See Figure 2.6 for an illustration of the subdomains. It follows that the extended solution
domain is given by
{ | }
L R
x D x L D (2.26)
The subdomains
L
and
R
can be viewed as two virtual and rigid surfaces, between which
the beam structure is installed. It is easy to see that during the time period 0 ( ) /
L n
t L D v ,
the platoon of n oscillators always stays in the extended solution domain , with the tail (the nth
oscillator) at the left end of
L
at the initial time (t = 0) and the head (the first oscillator) at the
right end of
R
at the final time ( ( ) /
L n
t L D v ). Within the ESD, the number of oscillators is
always n. It is in the ESD that the dynamic response of the coupled beam-oscillator system is of
interest. Therefore, a solution procedure that developed in the ESD does not need to check the
number of oscillators on the beam structure, and consequently, the four stages of beam-oscillators
interactions described in Section 2.2.1 can be treated seamlessly in .
24
2.3.3 Dynamic response by generalized assumed-mode method
As shown in Eqs. (2.5) and (2.8), the multi-span beam and moving oscillators are coupled by
the spring and damping forces at time-varying locations, which makes exact analytical solutions
impossible for this dynamic system. In this subsection, a state equation in the extended solution
domain is formulated by a generalized assumed-mode method. Afterward, the solution of the
state equation gives the dynamic response of the coupled beam-oscillator system.
Define an extended beam displacement ( , ) W x t in by
( , )
( , )
0
B
L R
x w x t
W x t
x
(2.27)
where ( , ) w x t is the transverse displacement of the beam structure governed by Eq. (2.5). Also,
define an extended eigenfunction ( ) x in the extended solution domain by
( )
( , )
0
B
L R
x x
x t
x
(2.28)
where ( ) x is a mode shape function of the beam structure given in Eq. (2.23). Let
( )
( )
k
x be
the kth extended eigenfunction defined by Eq. (2.28). With the above definitions, the extended
displacement ( , ) W x t is approximated by an m-term series
( ) ( )
1
( , ) ( ) ( ) ( ) ( ),
m
k k
k
W x t x q t x q t x
(2.29)
where
( )
( )
k
q x are generalized modal coordinates; ( ) x and ( ) q x are vectors of the form
(1) (2) ( )
(1) (2) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
m
T
m
x x x x
q t q t q t q t
⋯
⋯
(2.30)
Different from the conventional assumed-mode method that adopts admissible functions, the
exact eigenfunctions in the ESD are used in the series (2.29) so that all the boundary conditions
25
(essential and natural boundary conditions) and matching conditions are satisfied. These exact
eigenfunctions are comparison functions that can deliver more accurate and faster-converging
results, as shall be seen in the next section.
With the series (2.29), application of extended Hamilton's principle yields a discretized model
of the coupled system that is described by the matrix ordinary differential equation
1
1 2
0 0 ( ) ( ) ( )
0 ( ) ( ) ( )
T T
b s c b s c
g o c o c c o
M q t q t q t C C K K K
f M y t y t y t C C K K K
ɺɺ ɺ
ɺɺ ɺ
(2.31)
in which y(t) is a vector defined as
1 2
( ) ( ) ( ) ( )
T
n
y t y t y t y t ⋯ ;
b
M and
b
K are inertia and
stiffness matrices for the multi-span beam structure;
, , ,
1
, , ,
( ) ( ) ( )
( ) ( )
( ) ''( ) ''( )
'( ) '( )
T
b
T
p
t j s j s j
T
b
T
j
r j s j s j
M A x x x dx
k x x
K EI x x x dx
k x x
(2.32)
o
M ,
o
C and
o
K are inertia, damping and stiffness matrices for moving oscillators;
1 2
1 2
1 2
diag
diag
diag
o n
o n
o n
M m m m
C c c c
K k k k
⋯
⋯
⋯
(2.33)
1 2
, , , ,
s c s c c
C C K K K are coupling damping and stiffness matrices;
1 ,1
, ,
1
,
( )
( ) ( ),
( )
o
n
T
s i o i o i c
i
n o n
c x
C c x x C
c x
⋮ (2.34)
, , , ,
1
1 ,1 1 1 ,1
1 2
, ,
( ) '( ) ( ) ( )
( ) '( )
,
( ) '( )
n
T T
s i i o i o i i o i o i
i
o o
c c
n o n n n o n
K v c x x k x x
k x v c x
K K
k x v c x
⋮ ⋮
(2.35)
26
and
1 2
T
g n
f m g m g m g ⋯ is external force vector.
Now, by defining a state vector
2( )
( )
( )
( )
( )
( )
m n
q t
y t
z t
q t
y t
ℝ
ɺ
ɺ
(2.36)
convert Eq. (2.31) into an equivalent state equation as follows
( ) ( ) z t A t z b ɺ (2.37)
where
1 1
1 1
1 1
1
1
1 1
1 2
0 I
( ) ,
0
( )
,
( )
T
b s b c
o c o o
T
b b s b c
o g
o c c o o
M C M C
A t Q
P Q M C M C
M K K M K
P b
M f
M K K M K
(2.38)
with I being the identity matrix. Note that matrix ( ) A t is a function of time t because matrices
1 2
, , , ,
s c s c c
C C K K K depend on the oscillator locations
, o i
x , which are functions of t by Eq. (2.2).
The initial condition of Eq. (2.37) is
(0) (0) (0) (0) (0)
T
T T T T
z q y q y
ɺ ɺ (2.39)
with
0 0
(0) ( ) ( ) ( ) , (0) ( ) ( ) ( )
T T
q A x u x x dx q A x v x x dx
ɺ (2.40)
The state equation (2.37) with the initial condition (2.39) can be solved via numerical
integration.
In summary, the proposed method takes the following four steps to determine the dynamic
response of the coupled beam-oscillator system.
27
Step 1. Obtain the analytical eigenfunctions (mode shapes) of the multi-span beam structure by
the DTFM as described in Section 2.3.1;
Step 2. Define an ESD by the formulas in Section 2.3.2;
Step 3. In the ESD, establish a state equation through the application of the generalized
assumed-mode method, as described in the current section; and
Step 4. Solve the state equation via numerical integration by other means.
It should be emphasized that the number of oscillators in the above formulation is always a
constant n. As such, there is no need to check the number of oscillators on the beam structure for
the solution of the state equation (2.37). Neither is required to change or adjust algorithms during
the solution process, which is a must in a conventional solution method. Because of this, a standard
numerical integration technique, such as the Runge-Kutta method, can be easily implemented.
The proposed modeling and solution method can deal with an arbitrary number of moving
oscillators that have different speeds and initial inter-distances. Although constant speeds of
moving oscillators have been assumed in this work, oscillators with time-varying speeds in
acceleration and deceleration can be handled similarly without much technical difficulty.
2.4 Numerical Examples
The proposed modeling and solution methods are illustrated on a four-span beam with fixed-
fixed ends carrying three moving oscillators, as shown in Figure 2.7. In Section 2.4.1 the transient
response of the coupled system is computed, in Section 2.4.2 the convergence of the proposed
method is investigated, and in Section 2.4.3 the results obtained by the proposed method are
compared with those by the finite element method (FEM).
For numerical simulation, the parameters of the coupled system are chosen as follows.
Beam structure:
28
3 3 2 4
1 2 3 4
7.83 10 kg/m , =200 Gpa, =0.01 m , =0.05 m
200 m, 50 m
l l l l
E A I
L L L L L
Column supports:
2 4
, , ,
29 Gpa, 0.25 m , 0.05 m , 5 m
s j s j s j j
E A I h
Moving oscillators:
7
500 kg, 2 10 N/m, 0
i i i
m k c
Figure 2.7 Schematic of a four-span-beam carrying three moving oscillators.
Here it is assumed that all the column supports and oscillators have the same parameters. The
equivalent spring coefficients of each column support by Eq. (2.4) are obtained as
9 8
, ,
1.45 10 N/m, 2.9 10 Nm
t j r j
k k .
In this study, the fourth order Runge-Kutta method is used to solve the state equation (2.37).
To avoid errors accumulation in numerical integration, the step size in the Runge-Kutta method
must be carefully selected. To this end, the step size is given by
max
1 2
( ) t
r
(2.41)
where r is a positive constant, which shall be called step size ratio, and
max 1 1 2 2
max( , / , / ,..., / )
m n n
k m k m k m (2.42)
29
where
max
is the highest natural frequency of the beam structure selected in the assumed-mode
method, as shown in Eq. (2.29). Commonly, r is at least six to warrant accurate numerical results.
For the subsequent transient response simulations, the step size ratio is set as r = 10.
In the determination of the transient response of the coupled beam-oscillator system, 32 terms
are used in the series (2.29). The first 32 natural frequencies of the multi-span beam are determined
by the DTFM and they are listed in Table 2.1.
Table 2.1 Natural frequencies of the multi-span beam (rad/s).
Mode
Natural
frequency
Mode
Natural
frequency
1 56.855620 17 1128.885278
2 72.691873 18 1141.090880
3 91.017042 19 1231.844849
4 100.416247 20 1231.845054
5 198.150503 21 1518.390905
6 227.618734 22 1569.324564
7 259.088121 23 1699.473830
8 273.059292 24 1708.907441
9 427.004921 25 1960.616805
10 467.845960 26 2074.300499
11 510.761013 27 2240.733728
12 524.060314 28 2287.120423
13 740.721733 29 2516.490960
14 783.512472 30 2676.162586
15 837.542332 31 2869.359142
16 842.336815 32 2961.527102
2.4.1 Transient response
First, consider three oscillators with the same traveling speed 100 m/s
i
v . Assume that the
oscillators are equally spaced initially; that is,
0 0
1 2
1000 m d d . Because the length of the beam
structure is 200 m, it takes two seconds for an oscillator to travel from the left end to the right end
of the beam structure. Also, because the initial inter-distance between two adjacent oscillators is
30
1000 m, it takes 22 seconds for all the three oscillators to move to the right extended domain
R
D .
During this period, the beam carries either one oscillator or none.
By the proposed method, the spatial profiles of the transverse displacement of the four-span
beam are plotted in Figure 2.8 at 12 different times (t = 0.25, 0.75, 1.25, 1.75, 10.25, 10.75, 11.25,
11.75, 20.25, 20.75, 21.25 and 21.75 s), which form three clusters of times, with one cluster related
to the passage of an oscillator over the beam structure (e.g., 0 2 s t , 10 12 s t and
20 22 s t ). The beam displacement profiles in three clusters of times look quite similar because
it takes a relatively long time (8 seconds) for an oscillator to arrive at the left end of the beam after
a previous oscillator leaves the right end.
Figure 2.8 Spatial distribution of displacement of the four-span beam with three oscillators at
speed 100 m/s.
31
Figure 2.9 Beam displacement at the mid-span points with three oscillators at speed 100 m/s.
Figure 2.10 Spatial distribution of displacement of the four-span beam with three oscillators at
speed 150 m/s.
32
Figure 2.11 Beam displacement at the mid-span points with three oscillators at speed 150 m/s.
The transient response at the midpoint of each span of the beam structure (x = 25, 75, 125 and
175 m) is also plotted against time in Figure 2.9. As shown in the figure, the beam displacement
becomes significantly large when an oscillator is on the beam. It is seen from Figs. 2.8 and 2.9 that
the beam vibration excited by each oscillator is similar in pattern and amplitude. This is expected
because the oscillators travel at a relatively low speed, which results in a beam deflection similar
to that due to moving force (gravity).
The above-mentioned similar patterns of the oscillator-induced vibration can be altered if the
oscillator speed is increased. Figure 2.10 and 2.11 show the transient response of the beam
structure with the oscillators moving at 150 m/s, with the same setting as for Figs. 2.8 and 2.9.
From Figure 2.10, the patterns of vibration in the three clusters of times become obviously different.
Furthermore, through comparison of Figure 2.9 and Figure 2.11, it is seen that the oscillators
33
moving at 150 m/s cause larger vibration amplitudes, exhibiting the dynamic effects of the
oscillators moving on the beam vibration, at a higher speed.
Figure 2.12 Beam displacement at the mid-span points with 10 oscillators at speed 200 m/s.
Figure 2.13 Comparison of the beam displacement at x = 75 m: solid line-under moving
oscillators; dashed line-under moving forces.
34
Now consider a platoon of 10 oscillators that have the same traveling speed, but different initial
inter-distances. The initial locations of the oscillators are given by
1 2 3 4 5
6 7 8 9 10
0, 100 m, 400 m, 1000 m, 1050 m
1100 m, 1200 m, 1500 m, 1550 m, 1700 m
where the coordinates
i
are given in Eq. (2.1). With this setup, it can be shown that the number
of oscillators traveling on the beam structure has four possibilities: zero, one, two and three. This
time-varying number of oscillators makes the description of the beam-oscillator interactions quite
complicated, and thus renders the determination of the response of the coupled dynamic system
by a conventional method very difficult, if not impossible.
By the proposed method, however, the interactions between the beam structure and moving
oscillators are described in the ESD systematically; the dynamic response of the coupled system
can be determined conveniently and efficiently, without having to count the number of oscillators
on the beam structure at each moment. Shown in Figure 2.12 is the transient response of the beam
structure at the span midpoints, x=25,75,125, and 175 m. Because multiple oscillators are on the
beam structure from time to time, the vibration displacement patterns are quite different from those
in the three-oscillator case as shown in Figure 2.11. Also, the effects of moving inertia and spring
coupling on the beam response are investigated via consideration of the beam response under the
corresponding moving forces, which are equal to the gravitational forces of the oscillators and with
the same speed (200 m/s). The difference between the beam response at x=75 m under the moving
oscillators and that under moving forces is shown in Figure 2.13.
The numerical investigation reveals that the passage of many oscillators over the multi-span
beam may lead to an ever-increasing vibration amplitude of the structure. To show this, consider
a platoon of 10 oscillators with the same mass and spring coefficients as before. Let the oscillators
35
be equally spaced initially with an inter-distance of 1000 m. Assume that all the oscillators move
at the same speed of 800 m/s. Numerical simulations in the following two cases are performed:
Case (i) the oscillators have no dampers (c = 0); and
Case (ii) the oscillators have dampers of the same coefficient c = 20000 N s/m.
Figure 2.14 Beam displacement at x = 75 m, with 10 oscillators at speed 800 m/s:
(a) without damping (c = 0); (b) with damping (c = 20000 N s/m).
The transient response of the beam structure at x = 75 m is plotted in Figure 2.14 (a) for Case
(i) and in Figure 2.14 (b) for Case (ii). As can be seen from these figures, the vibration amplitudes
of the beam structure, without and with dampers, increase in a fashion like resonant vibration. This
indicates that the oscillators, having proper inter-distances and traveling over the multi-span beam
one by one, generate a periodic excitation that causes ever-increasing vibration amplitudes of the
structure. Through comparison of Figs. 2.14 (a) and (b), it is also seen that the inclusion of dampers
in the moving oscillators significantly reduce the beam vibration amplitudes. Nevertheless, the
dampers do not prevent the seemingly unbounded beam vibration amplitudes from happening
(because the beam structure has no damping). Furthermore, the beam displacements at the four
mid-span points (x = 25, 75, 125, 175 m), under the 10 damped oscillators, are plotted against time
36
in Figure 2.15, which all show this trend of ever-increasing vibration amplitude. This interesting
phenomenon deserves further research, which shall be demonstrated in chapter 4.
Figure 2.15 Beam displacement at the mid-span points, under 10 oscillators with c = 20000 N
s/m and at speed 800 m/s.
2.4.2 Convergence study
In this work, an assumed-mode method with extended eigenfunctions is used to determine the
dynamic response of the coupled beam-oscillator system. The number m of the terms in the series
(2.29) affects the accuracy of computation results. To see this, a convergence study is conducted.
For simplicity in comparison, the same four-span beam shown in Figure 2.7 is considered although
only one moving oscillator with speed v = 100 m/s is used. Consider the series (2.29) with m = 8,
16, 24, 32, 40 and 80 respectively. In the numerical simulations, the step size ratio in Eq. (2.41) is
set as r = 10. The transient response of the beam structure at the location x = 75 m is plotted in
Figure 2.16 and the displacement of the oscillator in Figure 2.17. For comparison of the dynamic
responses, the boxes marked in Figs. 2.16(a) and 2.17(a) are plotted in Figs. 2.16(b) and 2.17(b).
It is seen from these figures that the transient simulation results converge quickly as the number
37
of modes increases. Indeed, the difference between the transient responses with 32 and 80 modes
is quite small.
(a)
(b)
Figure 2.16 Beam displacement at x = 75 m, with one oscillator at speed 100 m/s:
(a) 0 2 s t ; (b) 1.6 1.85 s t .
38
(a)
(b)
Figure 2.17 Displacement of the oscillator:
(a) 0 2 s t ;(b) 0.95 1.05 s t .
39
To facilitate a quantitative measure of convergence, define a relative error E as follows
2
,
1
2
,
1
( )
( )
n
i ref i
i
n
ref i
i
f f
E
f
(2.43)
where n is the total number of temporal points generated in numerical simulation; f is an
approximate solution;
ref
f is a reference solution, and sub-index i stands for a solution
corresponding at a specified time
i
t . In this study, the solution obtained by 80 modes in the series
(2.29) is used as a reference solution.
Figure 2.18 Relative error of the beam displacement and oscillator displacement versus the
number of assumed modes.
The relative error E is plotted against the number m of modes in the assumed-mode method in
Figure 2.18, for the beam displacement at x = 25, 75, 125 and 175 m and with the number n of
temporal points in Eq. (2.43) being 200,001. It is seen from the figure that the relative error E
40
decreases monotonically as the number m of modes increases. Moreover, when m is equal to or
larger than 32, E is less than 0.5%. Thus, a 32-mode model of the beam structure is accurate enough
for the examples in Section 2.4.1.
2.4.3 Comparison with FEM
In this section, the proposed method is compared with the finite element method (FEM), in
terms of convergence, accuracy, and efficiency. The same coupled system (the four-span beam
with one moving oscillator at speed v = 100 m/s) in Section 2.4.2 is used. In the FEM discretization,
each of the four-span beam segments is divided into N elements, resulting in total 4N elements for
the entire structure. For these elements, Hermite polynomials are used as shape functions [57]. For
transient solutions, the discretized governing equations by the FEM are cast into a matrix state
equation, which is then solved by the fourth order Runge-Kutta method.
For comparison purposes, the proposed method with 40 and 80 modes and the FEM model with
40, 80, and 120 elements are considered in simulations. According to the convergence study as
shown in the previous section, the 80-mode assumed-mode model is used as a reference solution
herein. Plotted in Figure 2.19 are the time histories of the transient displacement of the beam
structure at x = 75 m, for 0 2 s t , which are generated by the proposed method with 40 and 80
modes and by the FEM with 40 and 120 elements, respectively. Note that the total travel time of
the oscillator over the beam structure is 2s. From Figure 2.19 (a), the predictions by the FEM and
the reference solution are in good agreement. A closer look at the transient response the time zone
of 0.749 0.754 s t in Figure 2.19 (b) shows the trend of the FEM solutions, which approach
the prediction by the reference solution as the number of elements increases. Furthermore, it is
seen that the proposed method with 80 modes is as good as the FEM with 120 elements.
41
(a)
(b)
Figure 2.19 Transient displacement of the structure at x = 75 m by the proposed solution and the
FEM : (a) 0 2 s t ; (b) 0.749 0.754 s t .
42
For evaluation of computational efficiency, the elapsed times in determination of the dynamic
response of the coupled beam-oscillator system by the proposed method and the FEM are listed in
Table 2.2. The simulations are run on a laptop computer with MS Windows, an Intel Core i7
processor, and 8GB memory, and with the step size ratio r in Eq. (2.41) being 10. It is seen from
the table that the proposed method saves tremendous computation time for the same accuracy. As
shown in Figure 2.19, the proposed method with 40 modes is more accurate than the FEM with 40
elements. However, the elapsed time by the proposed method is 72.2s, which is just 15.6% of the
time (462.7s) needed by the FEM. Savings in computation time become more significant when
higher accuracy of simulation is required. For instance, if an 80-mode model is used, which is as
accurate as the 120-element model (see Figure 2.19), the elapsed time by the proposed method is
merely 3.3% of the time needed by the FEM. Even if the proposed model with 80 modes is
compared with the FEM model with 80 elements, Table 2.2 shows a saving of 81.8% in
computation time by the proposed method. This computation efficiency of the proposed method
mainly comes from the usage of the exact eigensolutions of the beam structure that are provided
by the DTFM. On the other hand, the finite element modeling with a large number of elements
yields high-order matrices, which inevitably require significant effort and time in computation.
Table 2.2 Comparison of elapsed computation times.
Proposed
Method
(No. of Modes)
Elapsed Time
(second)
FEM
(No. of
Elements)
Elapsed Time
(second)
24 11.5 40 462.7
32 33.8 64 1377.8
40 72.2 80 4223.3
80 768.6 120 23062.3
43
It should be noted that the comparison herein only considers one moving oscillator. If multiple
moving oscillators are considered, a conventional FEM solution procedure must check the number
of oscillators on the beam structure at each computation step and make adjustment of the solution
algorithm accordingly. This issue of frequent check and adjustment makes the FEM solution
process tedious, complicated, and much more time-consuming. The proposed method, which is
established on the extended solution domain (ESD), totally avoids these issues and hence is
computationally more efficient.
2.5 Conclusion
A new technique for dynamic modeling and vibration analysis of a multi-span beam structure
carrying multiple moving oscillators has been presented. The main results from this chapter are
summarized as follows.
(1) An extended solution domain (ESD) is defined to facilitate a systematic and easy description
of beam-oscillator interactions. With the ESD, a generalized assumed-mode method is developed,
by which a set of time-varying state equations is obtained. The solution of the state equations by
numerical integration gives the dynamic response of the coupled beam-oscillator system.
(2) The highlight of this effort is the development of a capability of handling arbitrarily many
moving oscillators in modeling and analysis of the coupled-beam oscillator system, which is
unavailable with existing techniques. Because of the four stages of beam-oscillator interactions as
described in Section 2.2, a conventional solution method must check the number of oscillators
traveling on the beam structure at each computation step, and subsequently change the solution
algorithm among numerous possibilities. Due to this complication in modeling and solution, most
previous investigations have been limited to a few oscillators. On the other hand, the ESD-based
method does not need to do number checking on moving oscillators, and as such, it only requires
44
one algorithm (like standard numerical integration) in the entire solution process. This unique
feature allows the proposed method to determine the dynamic response of the coupled beam-
oscillator system with any number of moving oscillators.
(3) Different from the conventional assumed-mode method, the generalized assumed-mode
method adopts the exact analytical eigenfunctions of the multi-span beam in consideration, which
are delivered by the distributed transfer function method (DTFM). These exact eigenfunctions,
which satisfy all the boundary conditions of the beam structure and matching conditions at the
supports, are generally better than standard admissible functions. The generalized assumed-mode
method is thus highly accurate and efficient in computation, as shown in the numerical examples
in Section 2.4.
(4) The proposed method is illustrated on a four-span beam in two simulation cases: the
structure with three moving oscillators, and the structure with 10 moving oscillators. In the first
case, the accuracy, efficiency, and convergence of the proposed method in the computation are
fully demonstrated. In the second case, it is seen that a sequence of multiple moving oscillators
can excite beam vibration with ever-increasing amplitude. This interesting phenomenon, which is
somewhat similar to resonant vibration, deserves further research.
(5) The proposed method is compared to the finite element method (FEM) in a benchmark
problem of the four-span beam with one moving oscillator. It is shown that for the same accuracy
in simulation, the elapsed time required by the proposed method in computation can be less than
4% of the computation time needed by the FEM. This high numerical efficiency allows the
proposed method to model and simulate complicated coupled beam-oscillator systems.
Although the Euler-Bernoulli beam model has been used in this work, the proposed method can
be extended to other models of beam structures, including Timoshenko beam model and three-
45
dimensional beam model. Also, an elastic beam model for the supports can be introduced in the
formulation. Indeed, the exact eigenfunctions of a structure of more complicated configuration are
obtainable via the DTFM and the generalized assumed-mode method developed herein can be
applied.
46
Chapter 3 Beam-Moving Rigid Body Model
3.1 Introduction
The moving oscillator model is discussed in Chapter 2, where the elastic interaction between
the supporting structure and a moving subsystem is considered as spring and damping forces.
However, the dimensions or sizes of vehicles are ignored, and only one degree of freedom is
adopted in vehicle models. In many applications, consideration of vehicle sizes and multi-DOF
vehicle models is necessary to accurately investigate the dynamic response of a moving vehicle
and its effect on the structure. In recent years, rigid body vehicle models have been utilized by
researchers and various solution methods have been applied to compute the dynamic response for
the coupled system, e.g., modal superposition [43], Lagrange multiplier method [48], finite
element method [44,45,49], etc. Even though rich literature on the structure-moving system has
been done, continued investigation and efforts are still needed to get a deeper understanding of the
dynamic behavior for the coupled system. The previous research works are mainly limited to one
or just a few vehicles passing over the structure without considering a repeated passage and its
possible influence on the structure vibration. However, according to the author’s research on
dynamic analysis of structure-moving oscillator problem, the repeated passage of moving
oscillators could induce structure vibration with ever-increasing and significantly large amplitude.
This kind of dynamic behavior cannot be described by just a few vehicles and requires further
study.
In this chapter, a beam-moving rigid body model is proposed with which a moving subsystem
is simplified as a 2-DOF rigid body supported by two sets of spring-damper suspensions. Different
from an oscillator model, due to two contacting points between the structure and the rigid body,
even a single rigid body problem renders the dimension of associated mathematical model time-
47
varying when it is entering or leaving the structure. To describe the beam-rigid body interactions,
the extended solution domain (ESD) is applied here, where the number of degrees of freedom for
the associated mathematical model is determined and consistent, and the complicated coupling
between the structure and vehicles is systematically modeled. With the ESD, a set of ordinary
differential equations is generated with the help of distributed transfer function method (DTFM)
and generalized assumed-mode method whose solution gives the dynamic response of the coupled
system.
Additionally, parametric analysis is established to investigate the effect of system parameters
on the response of a structure carrying a sequence of moving rigid bodies. An ever-increasing (i.e.,
resonance-like) response of the structure is observed under certain circumstances if the velocity of
each vehicle and spacing distances between adjacent vehicles are chosen properly.
The remaining of this chapter is arranged as follows. In Section 3.2, the beam-moving rigid
body problem is described. The semi-analytical modeling and solution method is presented in
Section 3.3. Several extended cases are demonstrated in Section 3.4 in addition to the general case.
In Section 3.5, the dynamic response obtained with a moving rigid body model is validated by
comparing it to results obtained with FEM and moving oscillator model [58]. Effect of vehicle size
on the dynamic response of the structure is investigated by simulating cases with various vehicle
lengths. Parametric analysis of the vibration of the structure induced by passage of a sequence
rigid-bodies is studied in Section 3.5 as well. Conclusions are addressed in Section 3.6.
3.2 Problem Statement
A coupled structure-moving rigid bodies system in consideration is shown in Figure 3.1, where
a multi-span beam structure carrying multiple moving subsystems. The structure is modeled as a
stepped uniform Euler-Bernoulli beam supported by clamped or hinged boundary conditions on
48
left and right ends, and multiple elastic beam-columns in between. The moving subsystem is
modeled as a 2-DOF rigid body supported by two suspensions. The entire beam is divided into
(p+1) segments by the supports whose coordinates are denoted as
,1 ,2 ,
, ,...,
s s s p
x x x . For convenience,
x = 0 is set at the left end of the beam. n rigid bodies are moving consecutively over the structure
with constant velocities denoted as
i
v .
Figure 3.1 Schematic of the coupled beam-moving rigid bodies system.
Figure 3.2 Components of the coupled system:
(a) beam segment; (b) elastic column supports; (c) rigid body.
The coupled system has three basic components as shown in Figure 3.2: (a) beam segments,
with density
l
, Young’s modulus
l
E , cross-section area
l
A , area moment of inertia
l
I and length
l
L ; (b) elastic column supports with height
j
h , elastic modulus
, s j
E , cross-section area
, s j
A and
area moment of inertia
, s j
I ; (c) moving rigid bodies with mass
i
m , moment of inertia
, c i
I about
49
the center of mass, spring and damping coefficient
ij
k and
ij
c of the jth suspension with j=1, 2;
and total length
i
D , horizontal distance from the head to the center of mass
, G i
a , and horizontal
distance from the head to the jth suspension
ij
a being size parameters. Each rigid body has two
degrees of freedom, vertical displacement of the center of mass ( )
i
y t and rotation angle ( )
i
t .
3.2.1 Description of the passage of rigid bodies
Physically, the ends of the beam are connected to a left surface (x < 0) and a right surface (x >
L). Without loss of generality, assume that these neighboring surfaces are rigid ones without
deformation. In motion, a rigid body starts somewhere on the left surface, then travels on the beam,
and finally moves onto the right surface. When on the left surface, a rigid body is not coupled with
the beam. After entering the beam range, the rigid body is connected to the beam through its
suspensions, which induces vibrations in both the structure and the rigid body. When the rigid
body completely leaves the structure, its motion is uncoupled from the beam vibration again.
Figure 3.3 Initial locations and spacing distances of rigid bodies.
For the ith rigid body, denote the instant locations of the front (first) suspension of parameter
1 1
,
i i
k c and rear (second) suspensions of parameters
2 2
,
i i
k c by
1
( )
i
x t and
2
( )
i
x t which are functions
of time and
2
( )
i
x t can be calculated as
2 1 2 1
( ) ( ) ( )
i i i i
x t x t a a with
1 i
a and
2 i
a being the size
parameters shown in Figure 3.2. Assume at the initial time (t = 0), the front suspension of the first
50
rigid body is at the left end of the beam (
11
0 x ), and initial distances between adjacent rigid
bodies are
0
i
d defined as in Figure 3.3. Let the initial locations of suspensions of the ith rigid body
be
1 i
and
2 i
, then they are given by
11 12 12 11
1 1
0
1 1 11
1 1
1 1
0
2 2 11
1 1
0, ( )
( )
( )
i i
i i j k
j k
i i
i i j k
j k
a a
a a d D
a a d D
(3.1)
where 1, 2,..., i n and
k
D is the length of the kth rigid body as shown in Figure 3.3. For simplicity,
all the rigid bodies are moving at constant speeds, with the instant locations of front and rear
suspensions of the ith rigid body being calculated as
1 1 2 2 1 2 1
( ) , ( ) ( ) ( )
i i i i i i i i i
x t v t x t v t x t a a (3.2)
To avoid collisions, the inter-distance between any two adjacent rigid bodies is always positive.
In other words, the non-collision conditions can be specified as follows
0
1
( ) 0, [0, ], 1,2,..., 1
i i i f
d v v t t t i n
(3.3)
where
f
t is the total time for all the n rigid bodies to pass through the beam structure, and it is
2
( ) /
f n n
t L v .
It should be noted that the description of the passage of a sequence of moving rigid bodies over
a beam structure is much more complicated than that of a sequence of moving oscillators [39].
Unlike a moving oscillator, a moving rigid body has two suspensions. Because of this, a rigid body
is coupled with the beam if one of its suspensions is in the beam range. Indeed, there are three
cases of beam-rigid body coupling: (i) only the front suspension is coupled with the beam when
the body enters the beam range; (ii) both the front and rear suspensions are coupled with the beam
51
when the body travels within the beam range; and (iii) only the rear suspension is coupled with the
beam when the body is leaving the beam range. Because this complication in the description of
the beam-rigid body interactions, modeling and dynamic analysis of a beam structure carrying a
sequence of moving rigid bodies has not been well addressed in the literature. Due to the lack of
modeling and solution method for this problem, parametric vibration of the coupled system has
not been well investigated. The beam-rigid body interactions will be examined in detail in Sec.
3.3.2.
3.2.2 Governing equations of motion
In this work, elastic column support is modeled as a pair of translational and torsional springs.
The effective spring coefficients of the jth column support, according to structural mechanics [39],
are given by
, , , ,
, ,
,
s j s j s j s j
t j r j
j j
E A E I
k k
h h
(3.4)
where the geometric and material parameters are defined in Figure 3.2. With this simplification of
column supports, the coupled beam-rigid body system is governed by a partial differential equation
for the beam vibration and a set of ordinary differential equations for the motion of the moving
rigid bodies, which are presented as follows.
Beam structure:
2 2 2
2 2 2
( , ) ( , )
( ) ( )
s c
w x t w x t
A x EI x f f
t x x
(3.5)
where ( , ) w x t is the transverse displacement of the beam;
s
f and
c
f are the resultant forces due to
the elastic column supports and the structure-rigid body interaction respectively. The resultant
forces are of the form
52
, ,
, , , ,
1 1
( , ) ( )
( ) ( , ) ( )
p p
s j s j
s t j s j s j r j
j j
w x t d x x
f x k w x t x x k
x dx
(3.6)
,
2
1 1
,
[ ( ) ( , )]
( ) ( ) ( )
( ) ( , )
ij i G i ij i ij
n
c ij ij
i j
ij i G i ij i ij
k y a a w x t
f x x x t
D
c y a a w x t
Dt
ɺ
ɺ
(3.7)
where ( ) / D Dt is total differentiation, meaning ( , ) / / ( / ) ( / ) Dw x t Dt w t w x dx dt ; ( )
ij
t
are functions of time defined as ( ) ( ) ( )
ij ij ij
t h x h x L , with ( )
ij
x t being the instant location of
the jth suspension on the ith rigid body and ( ) h being Heaviside step function; ( ) is Dirac delta
function.
Moving rigid bodies:
2
,
1
2
,
1
[ ( ) ( , ) ( )]
( ) ( , ) ( )
i i ij i G i ij i ij ij
j
ij i G i ij i ij ij i
j
m y k y a a w x t t
D
c y a a w x t t m g
Dt
ɺɺ
ɺ
ɺ
(3.8)
2
, , ,
1
2
, ,
1
( )[ ( ) ( , ) ( )]
( ) ( ) ( , ) ( ) 0
c i i ij G i ij i G i ij i ij ij
j
ij G i ij i G i ij i ij ij
j
I k a a y a a w x t t
D
c a a y a a w x t t
Dt
ɺɺ
ɺ
ɺ
(3.9)
for 1, 2,..., i n . In addition, the boundary conditions for the beam with the fixed ends are
(0, ) '(0, ) 0
( , ) '( , ) 0
w t w t
w L t w L t
(3.10)
To fully describe the dynamic response of the coupled structure-vehicle system, the initial
conditions of the beam and moving rigid bodies are to be assigned. For simplicity in analysis, the
beam structure is assumed to have zero initial displacement that is measured from the equilibrium
configuration and zero initial velocity:
53
( ,0)
( ,0) 0, 0
w x
w x
t
(3.11)
Rigid bodies can have two types of initial conditions. In Case 1, the rigid body is initially at rest
in the vertical direction and the corresponding initial conditions are of the form
2 2
1 , 1 2 , 2
2
1 2 2 1
1 , 1 2 , 2
2
1 2 2 1
( ) ( )
(0) , (0) 0
( )
( ) ( )
(0) , (0) 0
( )
i G i i i G i i
i i i
i i i i
i G i i i G i i
i i i
i i i i
k a a k a a
y m g y
k k a a
k a a k a a
m g
k k a a
ɺ
ɺ
(3.12)
where gravity has been considered. This means that the body moves horizontally without
transverse motion until it enters the range of the beam. If the two suspensions are located
symmetrically about the center of mass with identical stiffness coefficient
i
k , then the initial
condition is reduced to
(0) , (0) 0, (0) (0) 0
2
i
i i i i
i
m g
y y
k
ɺ
ɺ (3.13)
In Case 2, the rigid body is subject to general initial disturbances, and as such, the body has
motion in the vertical direction before entering the beam range. The first case is useful in dynamic
analysis of the coupled system with the focus on solutions in the beam range. The second case, on
the other hand, can be used to investigate the dynamics of the coupled system in terms of “land-
bridge combination”. Either case of initial conditions can be easily implemented in the solution
process as to be presented in Sec. 3.3.
To determine the dynamic response, it requires to solve a mixed set of a partial differential
equation (3.5) and ordinary differential equations (3.8) and (3.9) simultaneously which are coupled
due to the resultant coupling force as shown in Eq. (3.7), and subject to boundary conditions (3.10)
and initial conditions (3.12). Solving this complicated set of equations with an analytical approach
54
is very difficult, if not impossible. A semi-analytical method is proposed in the next section that
provides a practical way to get the dynamic response of the coupled system.
3.3 Semi-Analytical Modeling and Solution Method
There are two key issues in solving the coupled dynamic system defined in Section 3.2. First,
the system is governed by a partial differential equation (3.5) and a set of ordinary equations (3.8)
and (3.9) which are coupled. To obtain the system response, they must be solved simultaneously.
Second, the interactions between the beam structure and moving rigid bodies make the system a
time-variant system, where a general analytical solution does not exist generally. Those issues
make it extremely difficult to apply conventional solution methods if not impossible. Thus, a semi-
analytical solution method is proposed here which is capable to handle those issues and provide
an efficient simulation algorithm to implement vibration and parametric analysis.
3.3.1 Analytical eigensolution for multi-span beam
In the proposed solution method, analytical eigenfunctions of the beam are used to model the
coupled beam-rigid body system. To this end, the distributed transfer function method is applied
[52,55,56,59].
For the lth segment of the beam structure, its eigenequation in the conventional sense is
4
2
4
( )
( ) 0, 0
l
l l l l l l
d u
Au E I L
dx
(3.14)
where is an eigenvalue (natural frequency) and ( )
l
u is the associated eigenfunction (mode
shape) on the lth segment; , , ,
l l l l
A E I and
l
L are the beam parameters defined in Sec. 3.2; and
is the local spatial coordinate defined on the segment. At the jth node, where the jth and (j+1)th
beam segments are interconnected and the jth elastic support is located, the matching conditions
are
55
1
, 1 1
, 1 1
( ) (0)
'( ) '(0)
'( ) ''( ) ''(0)
( ) '''( ) '''(0)
j j j
j j j
r j j j j j j j j j j
t j j j j j j j j j j
u L u
u L u
k u L E I u L E I u
k u L E I u L E I u
(3.15)
where ' / u du d . In Eq. (3.15), the first two equations are about displacement and slope
continuity and the last two are about force and moment balance.
By the DTFM, the eigenvalue problem of the beam structure (without moving rigid bodies) is
described by the equivalent spatial state equation
( ) ( ) ( ), 0
l l l l
d
F L
d
(3.16)
where ( )
l
is a state vector given by
( ) ( ) '( ) ''( ) '''( )
T
l l l l l
u u u u (3.17)
In terms of the state vector, the boundary conditions are written as
1 1 1
(0) ( ) 0
p p
M N L
(3.18)
and the matching conditions (3.15) is reduced to
1
(0) ( ), 1,2,...,
j j j j
T L j p
(3.19)
In Eq. (3.16), ( )
l
F is a four-by-four matrix containing the parameters for the lth beam segment
and ; M, N are boundary condition matrices and
j
T is matching condition matrix.
Recalling the formulation of the DTFM from Refs. 52 and 56, the analytical eigenfunction
associated with a uniform beam is expressed as
0
( )
Fx
x e , where
Fx
e is the fundamental matrix
and
0
is a constant vector determined by boundary conditions. The eigenvalue problem of the
beam is governed by ( ) / ( ) ( ) d x dx F x . Thus, the analytical eigenfunction in state form for
56
the entire multi-span beam denoted as ( ) x is expressed in terms of the eigensolutions of each
beam segment as
,1
1 1
, 1 , , 1
1 1
, ,
(0, ),
( )
( ) ( ) ( , ),
( )
( , ),
s
l l s l s l s l
p p
s p s p
x x x
x x x x x x
x x L x x
(3.20)
where ( )
l
x is a four-by-four matrix defined on the lth segment in exact analytical form as
2 2 2 2
3 3 3 3
cos( ) sin( )
sin( ) cos( )
( )
cos( ) sin( )
sin( ) cos( )
l
e e
e e
e e
e e
(3.21)
with being non-dimensional eigenvalue calculated as
2
4
/
l l l l
A E I and
l
is a
coefficient vector to be determined. The characteristic equation is defined as a transcendental
equation
0 1
1 1 1 2
1
1 1
(0)
( ) (0)
( ) det 0
( ) (0)
( )
p p p p
L p p
M
T L
T L
N L
⋱ (3.22)
where
0
M and
L
N are boundary condition matrices at left and right end respectively. The solution
to Eq. (3.22) gives the natural frequencies of the multi-span beam structure. The coefficient
l
is
determined by a non-trivial solution of the homogeneous equation (3.23).
0 1 1
1 1 1 2 2
1
1 1 1
(0)
( ) (0)
0
( ) (0)
( )
p p p p p
L p p p
M
T L
T L
N L
⋱ ⋮ (3.23)
57
With the coefficient vector
l
so determined, from Eqs. (3.20) and (3.21), the eigensolutions
of the multi-span beam are given in analytical and closed form. Mode shape and its spatial
derivatives are directly obtained from the state form eigenfunction ( ) x . This augmented
formulation of eigenfunction avoids using matrix exponentials that cause computational errors in
simulation and guarantees a bounded characteristic equation.
3.3.2 Extended solution domain
The fundamental problem of a beam structure coupled with multiple moving rigid bodies, as
defined in Sec. 3.2.1, is much more complicated than the problem of a beam carrying moving
oscillators. One major difference between these two problems lies in that an oscillator only has
one contact point while a rigid body has multiple contact points when interacting with a beam
structure. A moving oscillator is either on the beam or off the beam, which can be easily described.
For a rigid body with two suspensions as in Sec. 3.2, however, there are four cases of beam-rigid
body interactions: (1) the body is totally off the beam; (2) only the front suspension is on the beam;
(3) only the rear suspension is on the beam; and (4) both the front and rear suspensions are on the
beam. In the first case, there is no beam-body interaction and the motion of the body is not coupled
to the beam displacement. In the second and third cases, only one suspension of the body is in
touch with the beam, indicating that the body is entering or leaving the beam range, but it is not
completely on or off the structure yet. In the fourth case, the rigid body is completely on the beam
structure. If a rigid body has more than two suspensions or contact points, obviously there are more
cases of beam-body interactions.
The above-mentioned four cases make the description of beam-rigid body interactions
complicated with conventional methods, which is especially true when a sequence of multiple
moving rigid bodies is involved. Indeed, to precisely model such dynamic interactions, the number
58
of contact points between the beam structure and rigid bodies must be checked every moment.
Since these moving rigid bodies generally have different speeds and time-varying inter-distances,
detection of the number of contact points is complicated and tedious. This, in turn, renders
modeling and analysis of the coupled beam-rigid body system an extremely difficult task. And
because of this, many previous investigations have been limited to single or just a few rigid bodies
traveling within the beam range; that is, the fourth case of beam-rigid body interactions.
To deal with the complicated beam-rigid body interactions in modeling and analysis, the
extended solution domain (ESD) is applied here. It is an extension of the treatment of multiple
oscillators traveling over a beam structure in Chapter 2. One unique feature of the ESD is that the
number of degrees of freedom for the coupled system in the domain is a fixed number all the time.
The extended solution domain Ω for the coupled beam-rigid body system is the union of three
subdomains:
B L R
where the subdomains are as follows
Beam structure domain: { | 0 }
Left extended domain: { | 0}
Right extended domain: { | }
B
L L
R R
x x L
x D x
x L x L D
(3.24)
Figure 3.4 Schematic of extended solution domain for beam-rigid bodies system.
A schematic of the ESD is shown in Figure 3.4. Given n moving rigid bodies, the ESD is so
defined such that all the bodies stay within from the time when the first body enters the beam
59
range
B
and to the time the last body leaves
B
. Accordingly, the lengths of the left and right
extended domains are given by
1 1
0
2 11
1 1
1
( )
( )
n n
L n i i
i i
R L
n
D a a d D
v
D L D L
v
(3.25)
Geometrically,
L
D is the minimum length of the left extended domain to “park” all the n bodies
before the first one enters the beam range, and
R
D is the minimum length of the right extended
domain to “receive” all the bodies after the last one leaves the beam range.
The beam-rigid body interactions in the ESD are described as follows. The subdomains
L
and
R
are two virtual domains, which help obtain the vibration solutions of the coupled system.
Without loss of generality, the virtual domains are assumed as two rigid surfaces. (They can also
be assumed as deformable surfaces if necessary.) When a rigid body is moving in a virtual domain,
no interaction between the rigid body and the beam structure occurs, and the vertical motion of the
body is only influenced by its initial disturbance and gravity. If at least one suspension is in the
beam domain, the interaction between the beam structure and the rigid body takes place, coupling
the displacement of the beam structure to the motion of the rigid body.
The total time
f
t for n rigid bodies to move from the left extended domain
L
to the right
extended domain
R
is determined by ( ) /
f L n
t D L v . For 0
f
t t , these rigid bodies always
stay in the extended solution domain . In other words, during this period, the total number of
degrees of freedom of the coupled system in the ESD remains the same. Hence, in the ESD-based
modeling and analysis, checking on the number of contact points between the beam and moving
60
rigid bodies is not necessary, and solution algorithms for relevant dynamic problems become
straight-forward, as shall be seen in the next subsection.
3.3.3 Approximate model by generalized assumed-mode method
With the ESD, an extended displacement of the beam structure is defined by
( , )
( , )
0
B
L R
x w x t
W x t
x
(3.26)
where ( , ) w x t is the transverse displacement of the beam structure governed by Eq. (3.5). As
indicated by the previous equation, the beam structure is extended to the ESD, with zero
displacement in the virtual domains
L
and
R
for the rigid surface assumption.
According to Eqs. (3.5), (3.8) and (3.9), the multi-span beam structure and moving rigid bodies
are coupled by spring and damping forces at time-dependent locations. This makes the coupled
beam-rigid body system a time-varying dynamic system, for which analytical solutions are
extremely difficult to find, if not impossible. In this work, with the ESD, a generalized assumed-
mode method is applied to obtain the dynamic response of the coupled beam-rigid body system.
The proposed method is semi-analytical because it utilizes the analytical eigensolutions of the
beam structure as determined in Sec. 3.3.1.
In the generalized assumed-mode method, the extended displacement ( , ) W x t is approximated
by a truncated series as
( ) ( )
1
( , ) ( ) ( ) ( ) ( ),
m
k k
k
W x t x q t x q t x
(3.27)
where
( )
( )
k
x are extended eigenfunctions (mode shapes) and
( )
( )
k
q x are generalized coordinates;
( ) x and ( ) q x are vectors of the form
61
(1) (2) ( )
(1) (2) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
m
T
m
x x x x
q t q t q t q t
⋯
⋯
(3.28)
With the series (3.27), the kinetic and potential energy functionals of the coupled system can
be written. Then, the application of extended Hamilton’s principle yields the governing equation
for the coupled system in the following matrix form
1
1 2
0 0 ( ) ( ) ( )
0 ( ) ( ) ( )
T T
b s c b s c
g r c r c c r
M q t q t q t C C K K K
f M s t s t s t C C K K K
ɺɺ ɺ
ɺɺ ɺ
(3.29)
where s is a vector of the displacements and rotation angles of the rigid bodies and
g
f is a vector
of external forces, which are given by
1 1 2 2
1 2
0 0 0
T
n n
T
g n
s y y y
f m g m g m g
⋯
⋯
(3.30)
In Eq. (3.29),
r
M ,
r
C and
r
K are inertia, damping and stiffness matrices of moving rigid bodies,
given by
1 ,1 ,
1 1 ,1 1
2
1 ,1 1 1 ,1 1
2
1
,
2
, ,
1 1 ,1 1
2
1 ,1 1 1 ,1 1
,
diag
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )
r c n c n
j j G j
j G j j G j
r
j
nj nj G n nj
nj G n nj nj G n nj
j j G j
j G j j G j
r
nj nj G n nj
n
M m I m I
c c a a
c a a c a a
C
c c a a
c a a c a a
k k a a
k a a k a a
K
k k a a
k
⋯
⋱
⋱
2
1
2
, ,
( ) ( )
j
j G n nj nj G n nj
a a k a a
(3.31)
and
1 2
, , , ,
s c s c c
C C K K K are coupling damping and stiffness matrices due to beam-rigid body
interactions, given by;
62
2 2
1 1 1 1
( ) ( ), ( ) ( ) ( ) '( )
n n
T T T
s ij ij ij s ij ij ij i ij ij ij
i j i j
C c x x K k x x v c x x
2 2
1 1 1 1
1 1
2 2
1 ,1 1 1 1 ,1 1 1
1 1
1
2 2
1 1
2 2
, ,
1 1
( ) ( )
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( )
j j j j
j j
j G j j j G j j
j j
c c
nj nj nj nj
j j
nj G n nj nj nj G n nj nj
j j
c x k x
c a a x k a a x
C K
c x k x
c a a x k a a x
⋮ ⋮
2
1 1 1
1
2
1 1 ,1 1 1
1
2
2
1
2
,
1
'( )
( ) '( )
'( )
( ) '( )
j j
j
j G j j
j
c
nj n nj
j
nj n G n nj nj
j
c v x
c v a a x
K
c v x
c v a a x
⋮
(3.32)
Note that the coupling damping and stiffness matrices in Eq. (3.32) are all time-dependent
because the instant locations of suspensions associated with each rigid body ( )
ij
x t are functions
of time. This makes the coupled beam-rigid body system time-variant.
Now, define a state vector as
2 4
( )
( )
( )
( )
( )
m n
q t
s t
z t
q t
s t
ℝ
ɺ
ɺ
(3.33)
the governing equation (3.29) is converted into an equivalent state equation
63
( ) ( ) z t A t z b ɺ (3.34)
where
1 1
1 1
1 1
1
1
1 1
1 2
0 I
( ) ,
0
( )
,
( )
T
b s b c
r c r r
T
b b s b c
r g
r c c r r
M C M C
A t Q
P Q M C M C
M K K M K
P b
M f
M K K M K
(3.35)
with I being the identity matrix. Matrix ( ) A t is a function of time t because matrices
1 2
, , , ,
s c s c c
C C K K K depend on the rigid body locations ( )
ij
x t , which are functions of time by Eq.
(3.2). The initial condition of Eq. (3.34) is
(0) (0) (0) (0) (0)
T
T T T T
z q s q s
ɺ ɺ (3.36)
with
0 0
(0) ( ) ( ) ( ) , (0) ( ) ( ) ( )
T T
q A x u x x dx q A x v x x dx
ɺ (3.37)
where
0
( ) u x and
0
( ) v x are the initial profiles of the extended displacement and velocity of the
beam structure. Under the assumption that displacement of the beam is measured from its
equilibrium position with zero motion, the initial (0) (0) 0 q q ɺ .
3.3.4 Dynamic response of the coupled system
The number of degrees of freedom of the coupled system described by Eq. (3.29) is (m+2n),
with m being the number of terms in the eigenfunction series (3.27) and n being the total number
of the traveling rigid bodies. Because (m+2n) is fixed during 0
f
t t when the n rigid bodies
travel in the extended solution domain , an algorithm for solving Eq. (3.29) does not need to
check the number of contact points between the beam and rigid bodies, which otherwise is a must
with a conventional solution method. Therefore, the state equation Eq. (3.34) with the initial
64
condition (3.36) can be solved through use of conventional numerical integration methods, such
as the Runge-Kutta method or Newmark-beta method, which eventually yields the dynamic
response of the coupled beam-rigid body system.
In summary, the proposed semi-analytical method undertakes the following four steps to
determine the dynamic response of the coupled beam-rigid body system:
1. Obtain the analytical eigenfunctions of the multi-span beam structure as described in Sec.
3.31;
2. Define an extended solution domain (ESD) as described in Sec. 3.32; and
3. Construct the state equation (3.34) by the generalized assumed-mode method as described in
Sec. 3.33;
4. Solve the state equation for the dynamic response of the coupled system via numerical
integration.
3.4 Extension of the Proposed Method
In the previous section, the extended solution domain and generalized assumed-mode method
are utilized to determine the dynamic response of a beam structure carrying moving rigid bodies
of two contact points (suspensions) and with constant speeds. In this section, the proposed
modeling and analysis techniques are extended to more general coupled systems in three aspects:
(i) rigid bodies moving with time-varying speeds; (ii) rigid bodies with generally multiple contact
points (suspensions); and (iii) multi-span beam structures partially supported by viscoelastic
foundation. The extension should be useful in the determination of the dynamic response of more
complicated coupled systems, such as high-speed train systems and heavy trucks on highway
bridges.
65
3.4.1 Rigid bodies with time-varying speeds
The speed of a vehicle, in general, can be an arbitrary function of time, describing the physical
processes of its acceleration, deceleration, cruise control, and emergency braking. To reflect these
processes in the modeling of a coupled beam-rigid body system, time-varying speeds of rigid
bodies need to be implemented. To this end, denote the speed of the ith rigid body by ( )
i
v t , which
is a specified function of time. The instant locations of the two suspensions (contact points) of the
body are given by
1 1 2 2
0 0
( ) , ( ) ( )
t t
i i i i i i
x v t dt x t v t dt
(3.38)
with
ij
being the initial suspension locations as shown in Eq. (3.1). The non-collision condition
for the moving rigid bodies are specified as
0
1
0
[ ( ) ( )] 0, [0, ], 1,2,..., 1
t
i i i f
d v t v t dt t t i n
(3.39)
where
f
t is the total time of passage determined by
2
0
( )
f
t
n n
L v t dt
(3.40)
The integral in Eq. (3.38) can be evaluated by either analytical quadrature or numerical
quadrature. This can be done offline with high precision because the horizontal motion of a rigid
body is independent of its vertical motion. Thus, by writing
0
( ) ( )
t
i i
t v t dt
(3.41)
which is a known function of time, the instant suspension locations are expressed by
1 1 2 2
( ), ( ) ( )
i i i i i i
x t x t t (3.42)
It follows that the non-collision conditions (3.39) are reduced to
0
1
( ) ( ) 0, [0, ], 1, 2,..., 1
i i i f
d t t t t i n
(3.43)
66
with
f
t being calculated from
2
( )
n f n
t L .
Once the instant suspension locations (beam-body contact points) are determined, the extended
solution domain can be easily defined, with the lengths of the virtual domains being
2 2
11 11 1
(0)
( ) ( )
L n n
R f f
D x
D x t t
(3.44)
With the ESD, the generalized assumed-mode method can be applied to obtain a state equation of
the coupled system by following Sec. 3.33. As before, the solution of the state equation by
numerical integration gives the dynamic response of the coupled system.
3.4.2 Moving rigid bodies with multiple suspensions
A rigid body with two elastic suspensions is a typical half-model for automobiles but it may not
be enough to model train carriages or a heavy truck, which has more than two sets of wheels or
suspensions. In this section, moving rigid bodies with generally multiple suspensions (contact
points) are considered.
Figure 3.5 The ith rigid body with multiple suspensions.
Assume that n rigid bodies travel rightward through a beam structure. Shown in Figure 3.5 is
the ith rigid body with u suspensions, which are u pairs of springs and dampers of coefficients
ij
k
and
ij
c , with 1, 2,..., j u . In the figure,
, G i
a is the horizontal distance between the center of mass
67
and the right end of the rigid body;
ij
a is the horizontal distance between the ith suspension and
the right end;
i
m and
, c i
I are the mass of the body and the moment of inertia about its center of
mass. The rigid body has two degrees of freedom, vertical displacement of the center of mass ( )
i
y t
and rotation angle ( )
i
t .
For the rigid bodies as illustrated in Figure 3.5, an extended solution domain can be obtained
by following Sec. 3.32. With the ESD, application of the generalized assumed-mode method
produces a second-order matrix differential equation that has the same format as Eq. (3.29). The
inertia, damping and stiffness matrices in the equation are constructed as follows:
(i) The inertia matrices
b
M and
r
M , stiffness matrix of the beam
b
K and external force vector
g
f are the same as given in Sec. 3.33;
(ii) The damping and stiffness matrices of the rigid bodies are given by
1 1
,
r r
r r
n n
r r
C K
C K
C K
⋱ ⋱ (3.45)
where
,
1 1
2
, ,
1 1
,
1 1
2
, ,
1 1
( )
( ) ( )
( )
( ) ( )
u u
ij ij G i ij
j j
i
r
u u
ij G i ij ij G i ij
j j
u u
ij ij G i ij
j j
i
r
u u
ij G i ij ij G i ij
j j
c c a a
C
c a a c a a
k k a a
K
k a a k a a
(3.46)
(iii) The damping and stiffness matrices due to the beam-rigid body coupling are given by
68
1 1 1 1
( ) ( ), ( ) ( ) ( ) '( )
n u n u
T T T
s ij ij ij s ij ij ij ij i ij ij
i j i j
C c x x K k x x c v x x
1 1
1
1 ,1 1 1
1
1
,
1
( )
( ) ( )
( )
( ) ( )
u
j j
j
u
j G j j
j
c
u
nj nj
j
u
nj G n nj nj
j
c x
c a a x
C
c x
c a a x
⋮
1 1 1 1 1
1 1
1 ,1 1 1 1 1 ,1 1 1
1 1
1 1
1 1
, ,
1
( ) '( )
( ) ( ) ( ) '( )
,
( ) '( )
( ) ( ) ( ) '( )
u u
j j j j
j j
u u
j G j j j G j j
j j
c c
u u
nj nj nj n nj
j j
u
nj G n nj nj nj n G n nj nj
j j
k x c v x
k a a x c v a a x
K K
k x c v x
k a a x c v a a x
⋮ ⋮
1
u
(3.47)
It should be noted that the dynamic interactions between a beam structure and rigid bodies with
arbitrarily many contact points are much more complicated than those four cases presented in Sec.
3.2.1. Indeed, for traveling rigid bodies of multiple contact points, there may exist too many cases
of beam-body interactions to be handled by a conventional method. With the ESD and the
generalized assumed-mode method proposed in this paper, the issue of complicated beam-body
interactions is resolved, and the determination of the dynamic response of such a coupled system
does not require to check the number of beam-body contact points at all.
69
3.4.3 Beam structure partially supported by viscoelastic foundation
The beam structure considered in this section is generalized to include partially distributed
viscoelastic foundation, which may be used in more engineering applications. For instance, a
highway ramp that is partially laid on the ground can be modeled as a stepped beam supported by
both columns and partially distributed viscoelastic foundation.
For a beam segment with an elastic foundation, its eigenfunctions (mode shapes) are no longer
as simple as those of a beam without the foundation. Let the coefficient of the elastic foundation
on the lth beam segment be
, f l
k . Define a transition frequency as
,
/
tr f l l l
k A . Depending
on the value of
tr
, a mode shape function ( )
l
u of this beam segment takes the following two
different forms:
Case I:
2
4
,
, ( ) / 4
tr f l l l l l
k A E I
( ) ( )
( ) sin( ) sin( ) cos( ) cos( )
l l
L L
l l
u e e e e
(3.48)
Case II:
2
4
,
, ( ) /
tr l l f l l l
A k E I
( )
( ) cos( ) sin( )
l
L
l l
u e e
(3.49)
In Eqs. (3.48) and (3.49),
l
is a coefficient vector to be determined by the boundary conditions
and matching conditions of the beam structure as discussed in the Sec. 3.31.
By the DTFM formulation described previously, the eigensolutions of a multi-span beam
structure supported by a partially distributed elastic foundation can be similarly obtained.
Following the same process of the generalized assumed-mode method given in Sec. 3.33, the
dynamic response of the coupled system can be determined.
With the eigenfunctions given in Eqs. (48) and (49), a beam structure supported by a partially
distributed viscoelastic foundation can be easily modeled. This is done by adding a damping
70
coefficient of the foundation for each beam segment in the application of the generalized assumed-
mode method. The solution process for determination of the dynamic response of the coupled
beam-rigid body system is the same as before. Thus, the proposed modeling and solution method
is feasible for realistic applications on transport or rail systems.
3.5 Numerical Examples
The proposed modeling and solution method are illustrated in numerical simulation on single-
span beam and multi-span beam structures carrying moving rigid bodies. Also, the phenomenon
of parametric resonance is exhibited in the vibration simulation of a beam structure under a
sequence of moving rigid bodies.
3.5.1 A simply-supported beam carrying one vehicle
This example is used to validate the proposed method through comparison with the existing
results in the literature. Results obtained by the proposed method (with rigid body vehicle model)
are compared with those by a finite element method (with oscillator vehicle model) given in
reference [58]. Consider a simply supported beam with the following parameters: density per
volume
3
2400 kg/m , elastic modulus 27.5 Gpa E , cross-section area
2
2 m A , area
moment of inertia
4
0.12 m I , length 25 m L . The vehicle is modeled as a rigid body with two
degrees of freedom (Figure 3.2 (c)), with its parameters listed in Table 3.1, where two different
lengths of the rigid body are given. Let the vehicle speed be 10 m/s. Let the equilibrium position
of the vehicle be 0 y . The proposed method with 16 terms in Eq. (3.27) is compared with the
finite element method (FEM) [58] with 10 elements as a reference.
Table 3.1 Parameters of the vehicle.
Parameters Description Value
Mass 1200 kg
Spring coefficient 5 × 10
N/m
Damping coefficient 0 N ⋅ s/m
Mass 1200 kg
71
Moment of inertia 400 and 100 kg ⋅ m
Length 2 m and 1 m
Distance from head to the center of mass 1 m and 0.5 m
,
Spring coefficient of suspensions 2.5 × 10
N/m
,
Damping coefficient of suspensions 0 N ⋅ s/m
Distance from head to the front suspension 0.2 and 0.1 m
Distance from head to the rear suspension 1.8 and 0.9 m
The displacement of the vehicle at its mass center is plotted against time in Figure 3.6. It is seen
that the result by the proposed method with D = 1 m shows good agreement with the FEM reference
result. The result by the proposed method for D = 2 m, however, shows obvious deviations. This
is because the reference solution method [58] uses an oscillator model for the moving vehicle,
which overestimates the vibration of the vehicle at its mass center. Indeed, the dynamic response
of a rigid body passing over a beam structure can become significantly different from that of an
oscillator when its size cannot be neglected. In some engineering applications, a rigid body model
for moving vehicle becomes necessary in dynamic analysis for such coupled structure-moving
vehicle systems.
Figure 3.6 The displacement of the rigid body at its mass center:
− the proposed method (D=2m); -- the proposed method (D=1m); * the finite element method
given by Yang in Reference [58].
72
3.5.2 Effects of vehicle speed and size on the vibration of a four-span beam
It is well known that the dynamic response of a rigid body traveling over a beam structure can
be quite different from that of a moving oscillator, mainly because of the effects of the size and
rotational motion of the body. In this section, the effect of vehicle speed and size on the dynamic
response of a coupled system vibration is studied. The structure is a four-span beam with fixed-
fixed ends. The parameters of the structure are chosen as follows.
Beam structure:
3 3 2 4
1 2 3 4
7.83 10 kg/m , =200 Gpa, =0.01 m , =0.05 m
200 m, 50 m
l l l l
E A I
L L L L L
Column supports:
2 4
, , ,
29 Gpa, 0.25 m , 0.05 m , 5 m
s j s j s j j
E A I h
In the simulation, a single rigid body moves over the beam structure. There are four cases of
rigid bodies of different sizes; see Table 3.2 for the parameters of the rigid bodies. In each case, a
rigid body has different length D and moment of inertia
c
I about its mass center; the moment of
inertia
c
I is proportional to
2
D ; and other parameters, such as mass, spring coefficients, and
damping coefficients are the same.
Table 3.2 Parameters of rigid bodies of various lengths.
Parameters Case 1 Case 2 Case 3 Case 4
5000 kg 5000 kg 5000 kg 5000 kg
500 kg ⋅ m
12500 kg ⋅ m
5 × 10
kg ⋅ m
2 × 10
kg ⋅ m
1 m 5 m 10 m 20 m
0.5 m 2.5 m 5 m 10 m
,
2.5 × 10
N/m 2.5 × 10
N/m 2.5 × 10
N/m 2.5 × 10
N/m
,
0 N ⋅ s/m 0 N ⋅ s/m 0 N ⋅ s/m 0 N ⋅ s/m
0.1 m 0.5 m 1 m 2 m
0.9 m 4.5 m 9 m 18 m
73
The proposed method with 32 terms in Eq. (3.27) is used to compute the dynamic response of
the coupled system. The responses of the structure at the mid-point of the first and last beam spans
are computed for the vehicle speed v = 50, 100, 200 m/s and for four different rigid bodies as
described in Table 3.2. The computed beam responses are plotted against time in Figures. 3.7 to
3.9. The time interval in each of these figures are such that at the initial time (t = 0), the mass
center of the rigid body is at the left end of the structure (x = 0) and at the end of the time interval,
the mass center is at the right end (x = L). For instance, given the structure length of 200 m and for
a rigid body moving at speed v = 50 m/s, the end of the time interval is t = 4, as shown in Figure
3.7.
Figure 3.7 Comparison of dynamic responses for mid-points of beam spans
with rigid-body lengths being different values (v = 50 m/s).
74
Figure 3.8 Comparison of dynamic responses for mid-points of beam spans
with rigid-body lengths being different values (v = 100 m/s).
Figure 3.9 Comparison of dynamic responses for mid-points of beam spans
with rigid-body lengths being different values (v = 200 m/s).
75
At relatively low speed of the rigid bodies (50 m/s), the displacement of the beam structure is
dominated by static deformation and the interaction between the structure and the rigid body has
little contribution to the system response; as seen in Figure 3.7. At speed v = 100 m/s, the beam
responses show some small-amplitude oscillations at higher frequencies that are caused by the
interaction between the beam displacement and the rigid body motion, as observed from Figure
3.8. Once the speed becomes much higher (say 200 m/s), the contribution of the dynamic
interaction between the beam and the moving rigid body is greatly increased, and as a result, the
beam responses contain significant oscillations at higher frequencies; see Figure 3.9. This
phenomenon of ever-increasing coupling of beam displacement and vehicle motion at higher
speeds is also observed in an investigation on a beam structure coupled with moving oscillators
[39]. In Figure 3.10, rotation angle of the rigid body is plotted versus time for two speeds, (a) for
50 m/s and (b) for 200 m/s. The rigid body is having a more intense rotational vibration when it
moves at a higher speed. Thus, it is concluded that for a coupled beam-rigid body system, its
dynamic response cannot be accurately predicted just by using moving loads that are estimated by
a static deflection of the beam. The beam-rigid body interactions due to the beam flexibility and
the inertia of moving subsystems are non-negligible at a relatively high speed and can significantly
affect the response of the coupled system.
As seen from Figures. 3.7 to 3.9, a longer vehicle induces vibration in the beam structure with
a smaller amplitude and vice versa. For instance, the beam displacement induced by a 20 m rigid
body is much different from that induced by a 1 m rigid body. Two reasons may contribute to this
difference in induced beam displacements. First, upon entering the beam range, a longer vehicle
takes longer time to have its suspensions all on the structure, which results in different contact
forces. Second, a longer vehicle has a larger moment of inertia, by which, a bigger portion of
76
vibration energy goes in the rotational motion, thus reducing the amplitude of the translational
motion.
(a) (b)
Figure 3.10 Comparison of rigid body rotation angle with its length being different values:
(a) v = 50 m/s; (b) v = 200 m/s.
3.5.3 Three vehicles moving over a multi-span beam partially supported by viscoelastic
foundation
Dynamic interactions between a bean structure partially supported by a viscoelastic foundation
and moving rigid bodies have been discussed in Sec. 3.4.3. In this section, the effects of elastic
foundation on the dynamic response of a coupled system are investigated. To this end, consider
the same four-span beam structure as in Sec. 3.5.2. Let the first and last spans be supported by
elastic foundations with spring coefficient being
5 2
3 10 N/m and damping coefficient being
2
5000 Ns/m . Three identical rigid body vehicles with parameters given in Table 3.3 move over
the structure in sequence at the same speed. The inter-distance between two adjacent rigid bodies
is 200 m. 32 modes are used in the generalized assumed-mode method for simulation.
77
Table 3.3 The parameters of three identical rigid bodies.
Parameters Description Value
Mass 5000 kg
Moment of inertia 1 × 10
kg ⋅ m
Length 5 m
Distance from head to the center of mass 2.5 m
,
Spring coefficient of suspensions 2 × 10
N/m
,
Damping coefficient of suspensions 0 N ⋅ s/m
Distance from head to the front suspension 0.5 m
Distance from head to the rear suspension 4.5 m
Figures 3.11 and 3.12 show the transient responses of the structure (with the partially distributed
viscoelastic foundation) at the mid-span points (x = 25, 75, 125, 175m), which are induced by the
three rigid bodies traveling at the speeds 100 m/s and 200 m/s, respectively. At the lower speed
(100 m/s), the beam displacement excited by each rigid body has a similar pattern and amplitude.
This indicates that the beam vibration is mainly caused by the self-weight of the rigid bodies and
the effect of beam-rigid body interactions is less significant. At the higher speed (200 m/s), some
higher frequency vibrations show in the beam responses, which is caused by beam-rigid body
interactions.
If the viscoelastic foundation is removed from the structure, the beam responses induced by the
traveling rigid bodies have different patterns. Plotted in Figures. 3.13 and 3.14 are the beam
responses at the midspan points (x = 25, 75, 125, 175m), without the viscoelastic foundation. It is
observed from Figure 3.13 that at the speed of 100 m/s, small higher-frequency vibrations start to
appear in the beam responses, compared with Figure 3.11. At the speed of 200 m/s, the patterns of
the beam responses in Figure 3.14 are quite different from those in Figure 3.12, which again
exhibits the increased effects of dynamic interactions between the beam structure and moving rigid
bodies at a higher speed.
78
Figure 3.11 Response for mid-span points of the beam supported by viscoelastic foundation on
first and last spans with three rigid bodies moving over ( =100 m/s
i
v ).
Figure 3.12 Response for mid-span points of the beam supported by viscoelastic foundation on
first and last spans with three rigid bodies moving over ( =200 m/s
i
v ).
79
Figure 3.13 Response for mid-span points of the beam without viscoelastic foundation
with three rigid bodies moving over ( =100 m/s
i
v ).
Figure 3.14 Response for mid-span points of the beam without viscoelastic foundation
with three rigid bodies moving over ( =200 m/s
i
v ).
80
As indicated by Figures. 3.13 and 3.14, without a viscoelastic foundation, the vibration
amplitude of the beam structure can significantly increase at an increased speed of the moving
rigid bodies. A similar phenomenon was also seen in a moving oscillator problem [39]. These
results suggest that viscoelastic foundation could be used to reduce vibration of the beam induced
by moving rigid bodies. Therefore, optimal design and placement of the viscoelastic foundation
for a coupled beam-rigid body system is an interesting topic worthy of research effort.
3.5.4 Parametric resonance by a sequence of moving rigid bodies
A sequence of multiple vehicles traveling over a beam structure could induce vibrations with
ever-increasing amplitude. This phenomenon was observed in the author’s previous study of a
moving oscillator problem [39]. This kind of vibrations is caused by the dynamic interactions
between a flexible structure and periodically passing subsystems of inertia and spring components.
In other words, the interactions of a structure and moving subsystems could result in parametric
resonance, which can be of serious concern in relevant engineering applications.
In this section, with the proposed extended solution domain and generalized assumed-mode
method, the above-mentioned parametric resonance is demonstrated in a case study. The structure
in consideration is the same four-span beam as given in Sec. 3.5.2. A sequence of 10 rigid bodies
with equal inter-distance and same speed is considered. Each rigid body has identical parameters
that are as listed in Table 3.3. The spacing distance between two adjacent rigid bodies is 300 m. In
the numerical simulation, the first 32 modes in the series (3.27) are used. The natural frequencies
of those modes of the beam structure are given in in Table 3.4.
Table 3.4 First 32 natural frequencies of the four-span beam structure (rad/s).
Mode Natural
frequency
Mode Natural
frequency
1 56.855620 17 1128.885278
2 72.691873 18 1141.090880
3 91.017042 19 1231.844849
4 100.416247 20 1231.845054
81
5 198.150503 21 1518.390905
6 227.618734 22 1569.324564
7 259.088121 23 1699.473830
8 273.059292 24 1708.907441
9 427.004921 25 1960.616805
10 467.845960 26 2074.300499
11 510.761013 27 2240.733728
12 524.060314 28 2287.120423
13 740.721733 29 2516.490960
14 783.512472 30 2676.162586
15 837.542332 31 2869.359142
16 842.336815 32 2961.527102
Passage of a sequence of rigid bodies is somewhat like a periodic excitation that may induce
significantly large vibration of the beam structure. However, they are physically two different
types of excitations because the moving rigid bodies are involved in dynamic interactions with the
beam structure. The displacement of the beam structure at the midspan points (x = 25, 75, 125 and
175 m) are computed by the proposed method in the following three cases of the rigid body speed:
500 m/s, 1000 m/s and 1500 m/s. The results obtained are plotted in Figures. 3.15 to 3.17.
Figure 3.15 Beam displacement at x = 25, 75, 125 and 175 m induced
by rigid-bodies moving at 500 m/s.
82
Figure 3.16 Beam displacement at x = 25, 75, 125 and 175 m induced
by rigid-bodies moving at 1000 m/s.
Figure 3.17 Beam displacement at x = 25, 75, 125 and 175 m induced
by rigid-bodies moving at 1500 m/s.
83
It is seen from Figure 3.16 that the beam experiences vibration of ever-increasing amplitude. A
similar phenomenon has been observed in the vibration of a beam structure carrying a sequence of
moving oscillators [39]. This resonant vibration, however, is different from that of a time-invariant
system subject to harmonic excitation. The coupled beam-rigid body system is a time-variant
system and the passing rigid bodies involve in dynamic interactions with the flexible beam
structure. As seen from Figure 3.17, no obvious resonance-like response occurs at the speed of
1500 m/s. As for the case of 500 m/s (Figure 3.15), the beam displacement seems to increase in
the time of simulation but is not sure to increase unboundedly if more rigid bodies were to pass by.
All these indicate that the beam structure could be in parametric resonance.
The occurrence of the above-mentioned parametric resonance depends on the parameters of the
coupled system, and the corresponding stability conditions are yet to be determined in a follow-up
investigation. In this section, by computing the maximum beam displacement induced by a
sequence of moving rigid bodies at various speeds, a qualitative relation between the beam
displacement and the rigid body speed is established. In Figure 3.18, the maximum beam
displacement induced by the passage of 10 rigid bodies versus their speed is plotted. In the speed
range from 500 to 750 m/s, the maximum beam displacement is under 0.01 m, indicating that the
passage of 10 rigid bodies cannot induce a parametric resonance of the structure. However, at some
speeds, the passing rigid bodies excite the beam vibration with significantly large amplitude. At
these speeds, further numerical simulation shows that the beam vibration is ever-increasing along
with time if more moving rigid bodies are involved.
The effect of damping on the dynamic response of the coupled system is examined. In Figure
3.19, the maximum beam displacement versus the speed of the rigid bodies is plotted in three
damping cases: (i) the beam has no damping; (ii) the beam has 1% modal damping; and (iii) the
84
beam has 5% modal damping. As seen from the figure, modal damping can significantly reduce
the beam vibration amplitude, but it cannot completely remove those resonance peaks.
Figure 3.18 Maximum displacement of the beam induced by a sequence of 10 rigid bodies
moving at various speeds.
Figure 3.19 Maximum displacement of the beam with modal damping
induced by a sequence of 10 rigid bodies.
85
Figure 3.20 Maximum beam displacement induced by a sequence of 10 rigid bodies
with various inter-distances (770 m/s).
In Figure 3.20, the maximum beam displacement versus the spacing distance of the rigid bodies
is plotted at a speed of 770 m/s, which is a resonance peak in Figure 3.18. As shown in Figure 3.20,
the vehicular inter-distance significantly affects the beam response. The vehicular inter-distance
defines the period of passage of rigid bodies. A relation between the period and vehicle induced
resonance is an interesting and important topic of a future investigation.
To investigate the effects of vehicle models on the induced vibration of the beam structure,
three models are considered: (i) an oscillator model; (ii) a rigid body model with a length of 5 m;
and (ii) a rigid body model with a length of 20 m. The parameters of the 5m-long rigid body are
given in Table 3.2. The parameters of the oscillator and the 20m-long rigid body are given in Table
3.5. The same four-span beam structure without modal damping is used in the simulation and a
sequence of 10 vehicles with constant inter-distance as 300m are considered for all three models.
86
Table 3.5 Parameters of the vehicle for the oscillator and 20m-long rigid-body model.
Parameters Description Value
Mass 5000 kg
Spring coefficient 2 × 10
N/m
Damping coefficient 0 N ⋅ s/m
Mass 5000 kg
Moment of inertia 2 × 10
kg ⋅ m
Length 20 m
Distance from head to the center of mass 10 m
,
Spring coefficient of suspensions 2 × 10
N/m
,
Damping coefficient of suspensions 0 N ⋅ s/m
Distance from head to the front suspension 2 m
Distance from head to the rear suspension 18 m
Figure 3.21 Maximum beam displacement versus speed for three vehicle models.
The maximum beam displacements induced by three vehicle models are plotted against the
speed in Figure 3.21. All three models can induce parametric resonance in the beam structure. The
resonant frequencies are dependent on the size of a vehicle model: a longer rigid body model
having relatively higher resonant frequencies, and an oscillator model, which can be viewed as a
rigid body model of shrinking length, has lowest resonant frequencies. This may be explained by
87
the fact that it requires faster speed for vehicles to enter the beam domain at the same rate when it
has a longer length. However, the magnitude of the parametric resonance does vary for different
vehicle models which still need further investigation.
In summary of the numerical study presented in this section, two conclusions can be drawn.
First, a sequence of moving rigid bodies can induce parametric resonance of the supporting beam
structure through beam-moving body interactions. This kind of resonant vibrations cannot be
predicted by the traditional condition of an excitation frequency coinciding with a natural
frequency of the beam structure. Second, the occurrence of such parametric resonance depends on
the parameters of rigid bodies, including speed, size, and inter-distance. Derivation of analytical
conditions of parametric resonance for this type of coupled structure-vehicle systems is naturally
the next step in research, which is currently underway.
3.6 Conclusion
A semi-analytical method for modeling and dynamic analysis of a beam structure carrying
multiple moving rigid bodies has been developed. This method takes advantage of the exact
analytical eigenfunctions of the beam structure for high accuracy and efficiency in computation,
and it makes use of the extended solution domain (ESD) for the capability of describing complex
flexible structure-rigid body interactions in modeling and solution. The main results obtained in
this effort are summarized as follows.
(1) The dynamic response of a beam structure induced by the passage of a sequence of rigid
bodies is significantly different from that induced by moving forces. Due to the complexity of
dynamic interactions between the supporting beam structure and moving rigid bodies, the vibration
problem of such a coupled dynamic system has not been well addressed in the literature. In this
work, with the aid of the ESD technique, a new mathematical model is established for coupled
88
beam-moving rigid body systems. With the new model, the number of degrees of freedom of a
coupled system is fixed regardless of the number of bodies traveling on the beam structure at any
given time. This unique feature allows a simple and concise description of flexible-rigid body
interactions in modeling and easy implementation of numerical algorithms in solution.
(2) The proposed method is capable of modeling a variety of coupled beam-moving rigid body
systems, including a beam carrying multiple moving bodies, a multi-span beam carrying a
sequence of moving bodies, and a beam structure partially supported by a distributed viscoelastic
foundation and coupled with moving rigid bodies. While Euler-Bernoulli beam theory is used
throughout this work, the proposed method can be extended to Timoshenko beams and three-
dimensional beam structures. This modeling capability renders the proposed method versatile in
dealing with coupled systems in many engineering applications.
(3) The results from numerical simulations show different patterns of the dynamic response of
a coupled beam-moving rigid body system. When rigid bodies are moving at relatively slow speeds,
the dynamic response of the supporting beam structure is dominated by the static deflection caused
by the self-weight of the rigid bodies. In this case, the dynamic interactions between the flexible
structure and rigid bodies may be negligible, compared to the static deflection. When the speeds
of rigid bodies are high enough, however, the effects of dynamic interactions between the flexible
beam and rigid bodies become significant, and as such, the system response induced by the
interactions cannot be ignored. Moreover, while the placement of a viscoelastic foundation can
somewhat reduce the vibration amplitude of the beam, it cannot eliminate the higher frequency
response of the coupled system, which again is caused by flexible-rigid body interactions.
(4) A parameter study on vehicle-induced vibration reveals that the supporting beam structure
can experience parametric resonance when a sequence of rigid bodies passes over the structure.
89
Unlike traditional resonance induced by an external force with its excitation frequency matching
one natural frequency of the structure, the parametric resonance is caused by the periodic passage
of rigid bodies. This kind of phenomena is important in understanding the dynamic behaviors of
coupled systems in engineering applications, such as fast transportation and weaponry. Due to the
lack of a capable method in handling of complex dynamic interactions between the beam and
multiple moving rigid bodies, the parametric excitation problem for coupled beam-moving rigid
body systems has not been well investigated. The modeling and analysis method developed in this
work provides a platform for a thorough study of this problem.
(5) The numerical results also show that under certain conditions, the vibration of a supporting
beam structure induced by a sequence of moving rigid bodies can be significantly large, and it
cannot be eliminated with small damping in the structure. Apparently, further research on the
parametric resonance and derivation of relevant stability conditions is necessary.
90
Chapter 4 Parametric Resonance Analysis
4.1 Introduction
As shown in Chapter 2 and 3, a structure carrying a sequence of moving subsystems can have
ever-increasing vibration with significantly large amplitude. This kind of resonance-like vibration,
named parametric resonance, is caused by the repeated passage of subsystems, and highly depends
on parameters of the coupled system. For bridge-vehicle-interaction problem, the vehicle induced
resonance affects operation safety of the vehicle and comfort of passengers, causes rapidly
changed large stress in the structure which reduces the serviceability and life span of the bridge. It
is essentially important to have a deep understanding of parametric resonance and investigate the
influence of system parameters on the occurrence of the resonance.
Yang et al. [60] studied the resonance and cancellation condition of a simply supported beam,
by modeling the subsystems as a sequence of moving forces. With the assumed-mode method,
resonance of a single beam subject to boundary elastic supports induced by moving forces was
investigated by Yau et al. [61] and Yang [62] et al. They showed that the critical speed for the
cancellation to occur is independent of the support stiffness. Mao and Lu [63] found out that the
resonance severity is essentially governed by the ratio between the bridge and carriage lengths
with equally spaced moving load model. By using a moving force model for subsystems,
maximum resonance and resonant acceleration were studied by Museros [13] and cancellation
condition for an elastically supported beam was derived. Most of the research works are dealing
with moving force model since the analytical solution for beam-moving force problem is available.
However, the moving force model is limited in the application when the inertia of the moving
subsystem and elastic interaction between the structure and subsystems need to be considered.
Hence it is necessary to consider moving mass even moving oscillator or rigid body problem.
91
For moving mass, moving oscillator and moving rigid body problem, the key issue is that the
coupled system becomes time-variant due to the contact coupling between moving masses and the
structure, where a general analytical solution does not exist. Parametric excitation of a Timoshenko
beam due to passage of moving masses was studied by Pirmoradian et al. [64] with FEM
formulation and harmonic balance method. With the modal expansion method, Nikkhoo and
Rofooei studied the dynamics response of thin rectangular plates traversed by a moving mass [65]
and carried out parametric studies. It was extended to the case of sequential moving masses [66,67]
and the resonance velocity of masses is approximately predicted via evaluating the fundamental
frequency of the system. Most recently, resonance analysis of a straight beam [68,69] and a curved
beam [70] have been carried out by the application of FEM formulation with consideration of
complex vehicle model.
Among those studies, one major issue is that parametric analysis is achieved via numerical
methods, such as harmonic balance method or Newmark’s method. Through numerical integration
or iteration, dynamic response of the beam structure is computed for various parameters, which
delivers relations of resonance and system parameters. Additionally, for some research works, only
a few vehicles or carriages are considered in the simulation which does not exactly represent for
the parametric resonance induced by repeated passage of moving subsystems. The reason is
obvious because numerical simulation for a coupled system including a large number of moving
subsystems requires a lot of computation efforts and is very time-consuming. It is necessary to
have an analytical method that is capable to predict the resonance without doing numerous
simulations.
In this chapter, an analytical method is developed to predict the parametric resonance of
supporting structure induced by a sequence of moving subsystems. In the development, each
92
moving subsystem is modeled as a 1-DOF spring-mass-damper oscillator where the inertia effect
and flexible interactions are considered. The governing equation for the coupled system is derived
with the generalized assumed-mode method which has been demonstrated in the previous chapters.
Based on state equation, a mapping formulation is constructed to describe the dynamic response
for the coupled system. The resonance criterion is developed by evaluating the eigenvalues of the
mapping matrix which delivers an analytical prediction for the parametric resonance.
The remaining of this chapter are arranged as follows. In Section 4.2, the structure-moving
oscillator problem is described. The proposed modeling and solution methods are demonstrated in
Section 4.3. Numerical results about the parametric resonance are presented in Section 4.4 in terms
of parametric analysis about the system parameters on a benchmark problem. Conclusion and
discussion on future work are summarized in Section 4.5.
4.2 Problem Statement
The coupled system in consideration is shown in Figure 4.1, where the structure is a uniform
distributed continuum under transverse displacement ( , ) w x t , and each moving subsystem is a
spring-mass-damper oscillator with vertical displacement ( )
i
y t . For convenience, x = 0 is set at
the left end of the structure. n oscillators are moving consecutively over the structure with speed
i
v .
Figure 4.1 Schematic of the coupled structure-moving oscillator system.
93
4.2.1 Governing equations
Transverse displacement of the structure is governed by the following equation
, , ,
1
( , ) ( , ) ( , ) ( ) ( ),
n
tt t i i o i
i
Mw x t Dw x t Kw x t f t x x x
(4.1)
with boundary conditions
( , ) 0, , 1,2,...,
j b
w x t x j N (4.2)
and initial conditions
0 , 0
( ,0) ( ), ( ,0) ( ),
t
w x u x w x v x x (4.3)
where M, D, and K are spatial operators associated with inertia, damping, and stiffness;
,
( ) ( ) /
t
t and
,
( ) ( ) /
x
x represent partial derivative with respect to time variable t and
spatial variable x respectively;
i
f is coupling force between the ith oscillator and the structure;
,
( )
o i
x t is the instant location of the ith oscillator;
, ,
( ) ( ) ( )
i o i o i
t h x h x L is a unit pulse function
in terms of
,
( )
o i
x t ; ( ) is Dirac delta function;
j
are spatial operators on boundary ;
0
( ) u x
and
0
( ) v x are initial displacement and velocity of the structure. The coupling force is in the format
below where ( ) / D Dt represents total derivative with respect to time.
, , ,
[ ( , )] [ ( , )]
i i i o i i i t o i
D
f k y w x t c y w x t
Dt
(4.4)
Vertical displacement of each oscillator is governed by
, , , ,
[ ( , ) ] [ ( , ) ] 0, 1, 2,...,
i i tt i i t o i i i i o i i i
D
m y c y w x t k y w x t m g i n
Dt
(4.5)
with initial conditions
,
(0) , (0)
i i i t i
y a y b (4.6)
94
where , ,
i i i
m c k are inertia, damping coefficient and stiffness coefficient for the ith oscillator
respectively;
i
a and
i
b are initial displacement and velocity; g is the gravitational acceleration.
The coupled system is governed by a partial differential equation (4.1) and a set of ordinary
differential equations (4.5), subject to boundary conditions (4.2) and initial conditions (4.3) and
(4.6). Dynamic response of the coupled system requires solving Eqs. (4.1) and (4.5)
simultaneously. Since the instant locations of oscillators
,
( )
o i
x t are functions of time which render
time-dependent coupling forces
i
f , the coupled system is a non-autonomous (time-variant) system
where no general analytical solution can be obtained.
4.2.2 Description of periodic passage
In the previous chapters, the vibration of a beam structure with ever-increasing amplitude has
been observed when a sequence of oscillators or rigid bodies with identical parameters passing
over the structure at same constant speed and with equal spacing distances. Dynamic response of
the coupled time-variant system is different from a conventional time-invariant system subject to
external excitations. Nevertheless, the vibration of the structure is induced by the time-variant
flexible interactions between the structure and moving subsystems which is very like a periodic
excitation.
Generally, oscillators pass over the structure with various speeds and spacing distances. In this
section, the interest lies in the parametric resonance of the structure induced by a repeated and
periodic passage of oscillators. Without loss of generality, it is assumed that at any given time,
there is at most one oscillator moving over the structure ( , 1,2,..., 1
i
L d i n ), as shown in
Figure 4.2, where L is the length of the structure and
i
d represents the spacing distance between
the (i-1)th and ith oscillators. Any time when an oscillator enters the structure, the flexible
95
interaction between the structure and the oscillator starts to induce vibration of the structure. When
the oscillator leaves, the structure begins to have free vibration until the next oscillator arrives.
Figure 4.2 Schematic of the periodic passage of moving oscillators.
In previous chapters, a semi-analytical solution method that constituted of distributed transfer
function method (DTFM) and the generalized assumed-mode method was proposed, which could
handle the situation for an arbitrary number of moving subsystems. It was found that parametric
resonance could be induced in the structure if subsystems moving over the structure in a sequence
with certain patterns. However, this semi-analytical method can only simulate dynamic response
for the coupled system but not predict instability or parametric resonance with the system’s
parameters. The proposed method that developed based on mapping technique and capable to
achieve this goal shall be demonstrated in the sequel.
4.3 Modeling and Solution Methods
As discussed in previous chapters, the proposed semi-analytical method provides an effective
and efficient approach for dynamic analysis of the coupled beam-moving oscillator problem.
However, it cannot predict the parametric resonance with the system parameters, and it is
extremely time-consuming to perform parametric analysis using numerical integration. The novel
analytical method presented in this section is still based on the generalized assumed-mode method
96
with an augmented formulation which allows investigating into the mapping transformation of the
dynamic response of the coupled system between periods.
4.3.1 Generalized assumed-mode method formulation
The fundamental problem is to solve the coupled time-variant system. To this end, the same
semi-analytical formulation is applied here. The eigenvalue problem of the undamped system
associated with Eq. (4.1) is defined as
2
( ) ( ) 0,
k k
M K x x (4.7)
with boundary conditions
( ) 0, , 1, 2,...,
j k b
x x j N (4.8)
where
k
is the kth natural frequency of the undamped structure and ( )
k
x is the associated
eigenfunction (mode shape). Orthonormal condition of the eigenfunctions is defined as
, ( )
r k r k rk
M M dx
(4.9)
The analytical eigenfunctions of the continuum structure can be derived using augmented
DTFM formulation as demonstrated in Chapter 3. By the generalized assumed-mode method,
transverse displacement of the structure is approximated as a truncated series
1
( , ) ( ) ( ) ( ) ( )
m
k k
k
w x t x q t x q t
(4.10)
with ( )
k
q t being modal displacement associated with the kth mode; ( ) x and ( ) q x being vectors
in the form as
1 2
1 2
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
m
T
m
x x x x
q t q t q t q t
⋯
⋯
(4.11)
97
Consider modal damping with damping ratio
k
for the kth mode, the associated modal
displacement of the structure is governed by the following ordinary differential equation
, ,
1 2
, ,
1
, , , , , ,
1
( )[ ( ) ]
2 ( )
( )[ ( ( ) ( ) )]
m
i k o i i r o i r
n
r
k tt k k k t k k i
m
i
i k o i i t r o i r t i r x o i r
r
k x y x q
q q q t
c x y x q v x q
(4.12)
and displacement of each moving oscillator is governed by
, , , , , , ,
1 1
[ ( ( ) ( ) )] [ ( ) ] 0
m m
i i tt i i t i r o i r t i r x o i r i i i r o i r i
r r
m y c y x q v x q k y x q m g
(4.13)
Casting into matrix form, the governing equation for the coupled system is derived as
1
1 2
0 0 ( ) ( ) ( )
0 ( ) ( ) ( )
T T
b b s c b s c
g o c o c c o
M q t q t q t C C C K K K
f M y t y t y t C C K K K
ɺɺ ɺ
ɺɺ ɺ
(4.14)
in which y(t) is a vector defined as
1 2
( ) ( ) ( ) ( )
T
n
y t y t y t y t ⋯ ; ,
b b
M C and
b
K are inertia,
damping and stiffness matrices for the continuum structure;
1 1
2 2
1
I
diag 2 2
diag
b
b m m
b m
M
C
K
⋯
⋯
(4.15)
o
M ,
o
C and
o
K are inertia, damping and stiffness matrices for moving oscillators;
1 2
1 2
1 2
diag
diag
diag
o n
o n
o n
M m m m
C c c c
K k k k
⋯
⋯
⋯
(4.16)
1 2
, , , ,
s c s c c
C C K K K are coupling damping and stiffness matrices;
1 ,1
, ,
1
,
( )
( ) ( ),
( )
o
n
T
s i o i o i c
i
n o n
c x
C c x x C
c x
⋮ (4.17)
98
, , , ,
1
1 ,1 1 1 ,1
1 2
, ,
( ) '( ) ( ) ( )
( ) '( )
,
( ) '( )
n
T T
s i i o i o i i o i o i
i
o o
c c
n o n n n o n
K v c x x k x x
k x v c x
K K
k x v c x
⋮ ⋮
(4.18)
and
1 2
T
g n
f m g m g m g ⋯ is the external force vector. Solving Eq. (4.14) via
numerical integration gives the dynamic response of modal displacements for the structure and
oscillator displacements. However, since the system is time-variant, stability of the dynamic
response cannot be directly predicted by eigenvalues. To investigate the system stability, an
augmented formulation based on mapping transformation is developed which shall be
demonstrated in the next section.
4.3.2 Mapping transformation formulation
In section 4.2.2, an assumption is made that there is at most one oscillator moving over the
structure at any given time. Hence the time interval is divided into n segments as
0 1 2
0
n
t t t t ⋯
where
1 i
t
is the moment when the ith oscillator arrives at the left end (x = 0) of the structure and
n
t is the moment when the last oscillator reaches the right end (x = L). Thus, within an interval
1
[ , ]
i i
t t
, only the ith oscillator is considered being coupled with the extended structure defined in
an extended solution domain
i
as follows
1
| 0 ( )
i
i
t
i i
t
x x v t dt
(4.19)
For
1
[ , ]
i i
t t t
, the discretized governing equation for the coupled system that only includes the
ith oscillator is derived as
99
1
1 2
0
T T
b b s c b s c
i i i i i c i c c i
M q q q C C C K K K
m y y y m g C c K K k
ɺɺ ɺ
ɺɺ ɺ
(4.20)
where coupling submatrices are defined as
, , , , , ,
, 1 , 2 ,
( ) ( ) ( ), ( ) ( ) '( ) ( ) ( )
( ) ( ), ( ) ( ), ( ) '( )
T T T
s i o i o i s i i o i o i i o i o i
c i o i c i o i c i i o i
C t c x x K t v c x x k x x
C t c x K t k x K t v c x
(4.21)
By defining a state space vector [ ]
T T T
i b i
z z with [ ]
T T T
b
z q q ɺ and [ ]
T
i i i
y y ɺ , the
state equation is
1
( ) , [ , ]
i i i i
z A t z b t t t
ɺ (4.22)
where ( ) A t is a time-variant matrix and b represents the constant external force caused by gravity.
1 1
1 1 1
1
1
1 1
1 2
0 I 0 0
0
( ) ( )
0
( ) , 0 0 0 1
0
1 1
( )
m m m m
m T T
b b s b b s b c b c
m
m m
i i
c c c
i i i i
M K K M C C M K M C
A t b
k c
K K C
g
m m m m
(4.23)
The solution to Eq. (4.22) is in the form
1
1 1 1
( ) ( , ) ( ) ( , ) , [ , ]
i
t
i i i i i i
t
z t t t z t t bd t t t
(4.24)
where ( , ) t is the state transition matrix for ( ) z A t z ɺ calculated by the fundamental matrix
( ) t as
1
( , ) ( ) ( ) t t
. Rewriting the solution (4.24) with submatrices associated with
structure and oscillator separately yields the following expression
1
11 1 12 1 1 11 12
1
21 1 22 1 1 21 22
( ) ( , ) ( , ) ( ) ( , ) ( , )
, [ , ]
( ) ( , ) ( , ) ( ) ( , ) ( , ) i
t
b i i b i
i i
t
i i i i i
z t t t t t z t t t
bd t t t
t t t t t t t t
(4.25)
Consider the following circumstance:
All oscillators have identical parameters (inertia m, stiffness k, and damping coefficient c);
Oscillators are moving at the same constant speed v;
100
Spacing distances between any two adjacent oscillators are the same, denoted as d;
All oscillators are initially at equilibrium due to gravity with zero velocity.
A characteristic period is defined as / T d v and each interval becomes [( 1) , ] i T iT where
the ith oscillator is considered. The instant locations of oscillators satisfy the following relation
,
( ) [ ( 1) ] for [( 1) , ], 1,2,...,
o i
x t v t i T t i T iT i n (4.26)
which makes the coupling submatrices periodic as shown in Eq. (4.27). The time-variant matrix A
in the state equation (4.22) is also periodic with period T.
1 1 2 2
( ) ( ), ( ) ( )
( ) ( ), ( ) ( ), ( ) ( )
s s s s
c c c c c c
C t C t T K t K t T
C t C t T K t K t T K t K t T
(4.27)
From Floquet theory, the fundamental matrix for this problem satisfies the following relation
where C is a non-singular constant matrix.
( ) ( ) t T t C (4.28)
Periodicity of the state transition matrix can be easily proved from Eq. (4.28):
1 1 1 1
( , ) ( ) ( ) ( ) ( ) ( ) ( ) ( , ) t T T t T T t CC t t
(4.29)
Hence, Eq. (4.24) can be rewritten for each time interval as
11 12
21 22
11 12
( 1)
21 22
( ) [( 1) ] [ ,( 1) ] [ ,( 1) ]
( ) [( 1) ] [ ,( 1) ] [ ,( 1) ]
( , ) ( , )
, [( 1) , ], (
( , ) ( , )
b b
i i
t
i T
z t z i T t i T t i T
t i T t i T t i T
t t
bd t i T iT t t
t t
ɶ
ɶ ɶ
ɶ ɶ
ɶ ɶ
ɶ
ɶ ɶ
1) i T
(4.30)
Because all oscillators are at equilibrium position with zero vertical velocity initially, their
initial conditions at the start of each interval satisfy the following relation
0
[( 1) ] , 2,3,...,
i
i T i n (4.31)
Therefore, the transient response of the structure in terms of the modal displacement vector is
represented as the following for [( 1) , ] t i T iT
101
1
1 1
11 11 11
1
( ) ( ,0){ ( ,0) (0) ( ) ( ,0)} ( )
i
i j
b b
j
z t t T z T T t
ɶ ɶ
(4.32)
where ( ) t is external excitation as below
12 0 11 12
0
( ) ( ,0) ( , ) ( , )
t
t t t t bd
(4.33)
With a mapping matrix Z defined as
11
( ,0) Z T , stability criterion for the structure response
is stated in the following theorems:
Theorem 1: Modal displacement vector of the structure
b
z will be bounded if and only if
11
( ,0)
n
T and series
1
11
1
( ,0)
i
i
T
are bounded.
Theorem 2:
11
( ,0)
n
T is bounded and converges to zero if and only if ( ) 1 Z .
Theorem 3: The geometric series
1
11
1
( ,0)
i
i
T
is bounded and converges to
1
(I ) Z
if and
only if ( ) 1 Z .
( ) Z is the spectral radius of mapping matrix Z defined as ( ) max | |
i
Z . With the
theorems, stability of the structure response induced by a sequence of moving oscillators can be
easily determined by evaluating the eigenvalues of mapping matrix which is a submatrix of the
transition matrix for the coupled system. The transition matrix ( ,0) T for one period is calculated
as follows
1
( ,0) ( ) ( )
r
T T T ⋯ (4.34)
where ( )
i
T is the solution to a homogenous equation ( )
i i
A t ɺ and its initial value satisfies the
following relation with I
r r
being an identity matrix.
1
(0) (0) I
r r r
⋯ (4.35)
102
4.3.3 Highlights of the mapping transformation method
With the proposed mapping transformation method, an analytical stability criterion is
developed for the coupled structure-moving oscillator problem. Instead of tedious parametric
analysis using numerical simulation, the proposed method provides a very efficient approach for
predicting the parametric resonance of the structure induced by repeated passage of moving
oscillators. This novel method is capable to determine the resonance condition for the coupled
system by only evaluating spectral radius of the mapping matrix and predict the steady state
response caused by the periodic passage of moving oscillators as well which shall be shown sequel.
4.4 Numerical Results
In this section, numerical results about the parametric resonance are demonstrated. A
benchmark problem is presented in Section 4.4.1. Influence of the number of modes used in the
generalized assumed-mode method is studied and discussed in Section 4.4.2. In Section 4.4.3, a
parametric study is performed to get a deep understanding of parametric resonance of the coupled
structure-moving oscillator problem.
4.4.1 A benchmark problem for beam-moving oscillator system
In Section 4.2.1, a general formulation for the structure is defined using inertia, damping and
stiffness operators. At this point, a simply supported Euler-Bernoulli beam with zero initial
conditions is considered for simplicity with the following parameters:
4
50 kg/m, 200 Gpa, 0.05 m , 50 m A E I L
For a simply supported uniform Euler-Bernoulli beam, its normalized eigenfunctions are in the
form as
2
( ) sin( ), 0
k
k x
x x L
AL L
(4.36)
103
and the corresponding natural frequency is
2 2
2 k
k EI
L A
.
For most of the research works, only the first mode is considered [60-63] because the first mode
dominates the vibration of a simply supported beam. Hence only the first mode is considered in
the generalized assumed-mode method for this benchmark study.
Moving oscillators have identical inertia, damping and stiffness parameters as:
7
500 kg, 1 10 N/m, 0 m k c
Figure 4.3 Spectral radius of mapping matrix Z versus speed of oscillators (one mode
considered).
Moving speed of all oscillators are the same and the spacing distance between two adjacent
oscillators is chosen to be 130 m d so that the assumption that there is at most one oscillator
moving over the structure at any given time is satisfied. The parametric analysis is carried out for
the speed of moving oscillators to investigate the effect of oscillator speed on the parametric
104
resonance. The spectral radius of the mapping matrix Z is plotted versus the speed of moving
oscillators in Figure 4.3. Since only one mode is considered, the mapping matrix Z is a two-by-
two matrix with two eigenvalues.
According to Theorem 3, the spectral radius of Z is 1.011 when the speed of oscillators is 707
m/s, indicating a diverged and unbounded response of beam structure. To validate the conclusion
of resonance, transient response of the beam with 500 oscillators passing over is simulated.
Transient response of the mid-span point (x = 25 m) is plotted in Figure 4.4 (a) for 0 91.82 s t
and in Figure 4.4 (b) for 80 s 90 s t with a local scale. It is obvious that transient response of
the simply supported beam is diverged and unbounded which has been predicted by the spectral
radius of the mapping matrix.
(a)
105
(b)
Figure 4.4 Transient response of the beam at mid-span point (x = 25 m) for v = 707 m/s:
(a) 0 91.82 s t ; (b) 80 s 90 s t .
(a)
106
(b)
Figure 4.5 Transient response of the beam at mid-span point (x = 25 m) for v = 700 m/s:
(a) 0 92.74 s t ; (b) 85 s 90 s t .
Transient response of the beam induced by moving oscillators becomes totally different even
the speed of oscillators just deviates a little from the resonance speed (v = 707 m/s). The spectral
radius of the mapping matrix is 0.987 at v = 700 m/s, which indicates a stable and bounded beam
response. Transient response of the mid-span point (x = 25 m) is plotted in Figure 4.5 (a) for
0 92.74 s t and in Figure 4.5 (b) for 85 s 90 s t with a local scale. The displacement of the
simply supported beam has a beat-like transient vibration in the first 30 seconds and then converges
to the steady-state response after 60 seconds. It shows that the parametric resonance induced by
repeated passage of vehicles is very sensitive to system parameters.
107
Figure 4.6 Eigenvalues of series
1
n
i
i
Z
for v = 700 m/s.
The spectral radius of the mapping matrix Z is capable to predict not only the parametric
resonance but also the general trend of transient response for the beam structure. In Figure 4.6,
eigenvalues of the series
1
n
i
i
Z
are plotted versus the number of cycles n. As seen from Figure 4.5,
displacement of the beam has beat-phenomenon like transient vibration for the first 30 seconds.
This pattern can also be observed in the plot of eigenvalues of series
1
n
i
i
Z
as shown in Figure 4.6.
In this benchmark problem, the proposed mapping transformation method is validated by the
transient response of the simply supported beam. The resonance criterion derived from the
mapping matrix can successfully predict the parametric resonance without running the dynamic
simulation for the transient response. It is an extremely efficient approach for parametric analysis
108
for vehicle induced parametric resonance compared with traditional methods, like FEM [68-70].
With the proposed solution method, it is applicable to carry out parametric optimization for the
structure-moving vehicles system.
4.4.2 Effect of number of modes on parametric resonance criterion
In the previous section, a benchmark problem is studied to validate the proposed mapping
transformation method. Only the first mode is considered in the generalized assumed-mode
formulation for the parametric resonance analysis. As mentioned in Section 4.1, quite some
research work reduces the coupled structure-moving vehicles system into moving force problem,
where analytical solutions can be obtained. Besides eliminating the inertia effect of vehicles and
flexible structure-vehicle interactions, one major issue for the moving force model is that modes
of the beam structure are decoupled. Therefore, by applying sequential moving forces, each mode
of the structure will be excited independently. However, when a vehicle, say an oscillator, moving
over the structure, every mode of the structure is coupled with the vertical displacement of the
oscillator, and cannot be treated independently. In this section, the effect of the number of modes
on the parametric resonance condition is studied using the proposed mapping transformation
method.
The same simply supported beam and moving oscillators as in Section 4.4.1 are considered here.
To investigate the effect of the number of modes, five cases are considered for the simulation: 1,
2, 4, 6, and 8 modes. Apparently, the plot of spectral radius versus speed of oscillator differs for
each case significantly as shown in Figure 4.7. For the moving oscillator model, every mode is
coupled with the vertical displacement of the oscillator and the total contribution of all modes
results in the coupling force caused by flexible structure-vehicle interaction. Hence, for a more
109
realistic model where inertia effect and flexible structure-vehicle interaction are included, the
number of modes needs to be treated carefully.
(a)
(b)
(c)
110
(d)
(e)
Figure 4.7 Spectral radius of the mapping matrix for various numbers of modes:
(a) 1 mode; (b) 2 modes; (c) 4 modes; (d) 6 modes; (d) 8 modes.
When the number of modes considered in the generalized assume-mode method is increasing,
some peaks associated with resonance for a smaller number of modes are suppressed. Because all
modes are coupled with the displacement of the oscillator, the passage pattern which is associated
with resonance for one mode approximation does not induce the structure and generate parametric
resonance for cases with more modes. However, some peaks preserve for all cases as shown in
Figure 4.7 (d) and (e), which suggests that those resonance conditions are not affected by an
increasing number of modes in consideration.
111
This is only a simply supported beam structure where the first mode dominates the beam
vibration. Even that, the number of modes could influence the parametric resonance significantly.
The resonance conditions could be shifted or eliminated by taking more and more modes into
consideration. In order to get a reliable prediction on parametric resonance, a convergence study
of spectral radius need to be done.
4.4.3 Transient response of induced structure vibration
It has been shown that for a stable and bounded beam vibration, the steady-state response will
be reached once enough number of oscillators have passed by. In Figure 4.5, beam vibration has a
reducing magnitude for the case when oscillators are moving at v = 700 m/s. However, the transient
vibration of the beam structure can become very different if system parameters are different. In
this section, the generalized assumed-mode method with 6 modes is applied to the same simply
supported beam structure. Oscillators have identical parameters (inertia, stiffness and damping
coefficient) as shown in Section 4.4.1 and equal spacing distance 130 m d . Numerous cases are
simulated to investigate the transient response of the beam structure. The spectral radius of the
mapping matrix Z is plotted versus the speed of oscillators in Figure 4.8. Two resonance speeds
are marked in the figure, v = 643 and 906 m/s, and two bounded speeds are chosen as, v = 806 and
988 m/s.
With 500 oscillators sequence being considered, transient responses of the beam at the mid-
span point (x = 25 m) are plotted for four cases: 643 m/s in Figure 4.9, 906 m/s in Figure 4.10, 806
m/s in Figure 4.11 and 988 m/s in Figure 4.12. In Figures 4.10 and 4.11, parametric resonance
occurs which leads to an ever-increasing vibration. The divergence rate for the case of 906 m/s is
greater than that for the case of 643 m/s because the spectral radius of mapping matrix for the case
of 906 m/s is larger (1.015) than that of 643 m/s case (1.008).
112
Figure 4.8 Spectral radius of the mapping matrix versus speed of oscillators (6 modes
considered).
Figure 4.9 Transient response of mid-span point (x = 25 m) for v = 643 m/s.
113
Figure 4.10 Transient response of mid-span point (x = 25 m) for v = 906 m/s.
Figure 4.11 Transient response of mid-span point (x = 25 m) for v = 806 m/s.
114
Figure 4.12 Transient response of mid-span point (x = 25 m) for v = 988 m/s.
In Figures 4.11 and 4.12, two cases with stable and bounded vibration are presented. However,
it shows very different transient vibration response for those cases. In Figure 4.11, the transient
response of beam displacement at the mid-span point is relatively small and decreasing for the first
5 seconds, and then converge to its steady-state response. In Figure 4.12, transient response of
beam displacement is ever-increasing for the first 8 seconds and then converges to its steady-state
value which is relatively large compared with that for v = 866 m/s.
For 0 5 s t , transient response of mid-span point displacement is plotted for 806 m/s and
988 m/s respectively in Figure 4.13 (a) and (b). For 806 m/s case, the transient response has some
higher frequency vibration caused by flexible structure-vehicle interactions. In spite of that, the
beam has bounded and relatively small vibration (less than
3
3 10
). However, for the case of 988
m/s, the beam seems to have a diverged and unbounded vibration if only looking at the response
for first few seconds as shown in Figure 4.13 (b).
115
(a) (b)
Figure 4.13 Transient response of mid-span point displacement for 0 5 s t :
(a) v = 806 m/s; (b) v = 988 m/s.
This kind of different patterns in transient response for bounded vehicle induced structure
vibration validates the hypothesis that dynamic response of a structure induced by a sequence of
many vehicles could be significantly different from that induced by one or just a few vehicles. As
shown in Figure 4.13 (b), for the first 5 seconds when around 40 oscillators have passed by,
transient response of the beam is not sufficient to give a reliable prediction on the convergence of
beam vibration. What’s more, even for a bounded vibration as shown in Figure 4.12, it is still not
an optimal situation because of the large amplitude of its steady state response. With the
conventional numerical methods [68-70] for the coupled structure-moving vehicle problem, it only
considers a limited number of vehicles passing by which cannot predict the real dynamic behavior
for a structure induced by a sequence of moving vehicles. The proposed mapping transformation
method does not require tedious simulation for a large number of vehicles, renders a very simple
and efficient analytical resonance criterion.
116
4.4.4 Effect of modal damping on parametric resonance
Since the vehicle induced parametric resonance is essentially harmful to safety and lifetime of
a structure, effective vibration reduction is necessary for such systems. Because the parametric
resonance is different from the conventional resonance caused by periodic external excitations,
traditional vibration suppression may not be effective in this problem. In this section, a case study
for the effect of modal damping on the parametric resonance of a simply supported beam carrying
a sequence of moving oscillators is carried out. The parameters for the simply supported beam are
defined below:
4
50 kg/m, 200 Gpa, 0.05 m , 50 m A E I L
For the simulation, four modes are considered in the generalized assumed-mode formulation,
which renders an 8-by-8 mapping matrix. For the first case, parameters of oscillators are chosen
as:
7
500 kg, 1 10 N/m, 0 m k c . Natural frequency of the oscillator is 141.4214 rad/s and the
first two natural frequencies of the simply supported beam are 55.8309 rad/s and 223.3237 rad/s
which are calculated from Section 4.4.1. The natural frequency of the oscillator is chosen to be
between the first two natural frequencies of the beam. The spectral radius of the mapping matrix
is plotted versus speed of oscillators for two cases: (i) beam has zero modal damping; and (ii) beam
has 1% modal damping on each mode, in Figure 4.14 (a) and (b) respectively.
117
(a)
(b)
Figure 4.14 Spectral radius of mapping matrix (4 modes) versus speed of oscillators:
(a) with zero modal damping; (b) with 1% modal damping.
As shown in Figure 4.14, 1% modal damping pushes the curve of the spectral radius of mapping
matrix below one for all speeds between 50 and 1200 m/s and renders a much smoother spectral
radius history. According to the theorems presented in Section 4.3.2, no parametric resonance is
induced when 1% modal damping is added to the structure for all modes. For the 1% modal
damping case, even though the spectral radius is below 1.0 for all speed which indicates the
transient response of the beam structure is inside the stable region, the bumps and valleys of the
spectral history may represent some characters of transient response and deserve further
investigation.
In the second case, parameters of the oscillators are chosen as
7
500 kg, 2.5 10 N/m, 0 m k c
which makes its natural frequency 223.6068 rad/s that is very close to the second natural frequency
of the beam. The spectral radius of the mapping matrix is plotted versus speed of oscillators for
118
two cases: (i) beam has 1% modal damping; and (ii) beam has 5% modal damping to each mode,
in Figure 4.15 (a) and (b) respectively.
(a)
(b)
Figure 4.15 Spectral radius of mapping matrix (4 modes) versus speed of oscillators:
(a) with 1% modal damping; (b) with 5% modal damping.
From Figure 4.15(a), it is seen that there is a bump in spectral radius profile around 1050 m/s
that is associated with parametric resonance because the spectral radius is greater than 1.0. In other
words, with 1% modal damping, the beam could still have parametric resonance for certain speed
of oscillators. And it is an interval of speed that could induce parametric resonance instead of a
single speed as for a lumped system. In Figure 4.15 (b), the spectral radius is strictly less than 1.0
119
when 5% modal damping is added to every mode of the beam, indicating no parametric resonance
can be induced in the structure.
Indeed, modal damping has a positive effect on vibration reduction for parametric resonance.
It can improve the dynamic performance of the structure. However, suppression of parametric
resonance cannot always be achieved by only adding modal damping because the parametric
resonance is very sensitive to system parameters. Additionally, for real structures, large modal
damping is difficult to realize, and viscoelastic foundation can only be applied at the supports for
elevated multi-span structure.
4.5 Discussion and Conclusion
4.5.1 Extension of the proposed method
As demonstrated in Section 4.3, the proposed method is formulated based on the following
assumptions:
There is at most one oscillator moving over the structure at any given time;
All oscillators have identical parameters (inertia, stiffness and damping coefficient);
Oscillators are moving at the same constant speed;
Spacing distances between adjacent two oscillators are the same.
Those assumptions deliver a neat analytical formulation for demonstration purpose of the
methodology. Generally, more complex cases can be handled systematically without difficulties.
There are two extensions discussed here: (i) multiple oscillators are moving over the structure at
given any given time; (ii) clusters of vehicles including oscillators and rigid bodies are passing by
the structure sequentially.
For the first extension case, a schematic is shown in Figure 4.16. All oscillators are still assumed
to have identical parameters and moving at the same constant speed. The spacing distance d
120
between two adjacent oscillators satisfied the relation 1 / r L d r with r being an integer. This
model demonstrates the situation when a train carriage or a long truck passing by the structure.
Figure 4.16 Schematic of beam structure with multiple oscillators passing over.
For the second case, consider a sequence of cluster of oscillators or rigid bodies, as shown in
Figure 4.17. For one cluster, system parameters and moving speeds of the oscillators and rigid
bodies can be different. Parametric resonance can be induced by repeated passage of identical
clusters. With the mapping transformation method derived in this chapter, constant mapping
matrix can be obtained using numerical evaluation for one period between two clusters and the
spectral radius of mapping matrix shall be used to predict the parametric resonance. This case
represents a general description for structure-moving vehicle problem.
Figure 4.17 Schematic of beam structure carrying clusters of moving oscillators.
121
4.5.2 Conclusion
A semi-analytical method has been developed for parametric resonance prediction for coupled
structure-moving vehicle problem. This method is derived based on the semi-analytical solution
method for the structure-moving subsystem problem, which is developed in previous chapters, and
mapping transformation based on Floquet theory. The main contributions of this study are
summarized as follows.
(1) Dynamic response of a beam structure induced by passage of a sequence of vehicles is
significantly different from that induced by one or just a few ones. Parametric resonance can be
induced in the structure which is different from conventional resonance defined for a lumped
system. Since the coupled system is a non-autonomous or time-variant system, no general
analytical solutions are available. In this work, a semi-analytical method is developed which is
capable to predict the parametric resonance induced by a sequence of oscillators. Compared with
conventional methods where moving vehicles are simplified as moving concentrated forces, the
proposed method is derived based on the moving oscillator model that includes the inertia effect
and flexible structure-vehicle interactions.
(2) The proposed method is the first to provide an analytical and exact resonance criterion for
a structure-moving vehicle problem with moving oscillator model. Existing methods are either
using numerical evaluation with various system parameters for moving oscillator or moving rigid
body model, which is tedious and time-consuming or reducing the system to moving force problem
where inertia effect and flexible interaction are neglected. Applying the mapping transformation,
the proposed method delivers the analytical resonance criterion by evaluating the eigenvalues of a
constant mapping matrix without doing tedious simulation for dynamic response of the system. It
122
is extremely efficient and can be easily implemented into an optimization algorithm for design
purpose.
(3) A benchmark problem is studied for a simply supported beam carrying moving oscillators.
Numerical results validate the prediction from the proposed method that parametric resonance
occurs when the spectral radius of the mapping matrix is greater than one. And it also shows that
the parametric resonance is very sensitive to system parameters and could be affected significantly
even the parameters are only shifted a little.
(4) Numerical results show that the number of modes used in the generalized assumed-mode
formulation has a significant influence on the spectral radius profile. Because all modes are
coupled with the displacement of the oscillator, even for a simply supported beam where the first
mode is dominated, the resonance condition can change if a different number of modes is
considered. To get an accurate evaluation of resonance condition, a convergence study is necessary
on the number of modes needed for the generalized assumed-mode formulation. Because of this,
prediction obtained with moving force model is not accurate since every mode is decoupled in the
modal expansion formulation for moving force problem.
(5) A preliminary study about the effect of modal damping on parametric resonance is carried
out. Numerical results show that modal damping is capable to improve the dynamic performance
of the structure to some extent. But it is impossible to suppress the parametric resonance by only
adding modal damping for some circumstances. Since the modal damping cannot be too large for
real structures, a more effective way of vibration suppression is required.
123
Chapter 5 Dynamic Analysis of a Fast Projection System
5.1 Introduction
The problem of dynamic response for a continuum structure carrying moving subsystems has a
lot of engineering applications. Examples are vast, including highway bridges, elevated railways,
weaponry projection system, crane structure, cable transportation, etc. The dynamic behavior of
such system can become significantly different from those of corresponding stationary structures,
especially when subsystems are under extremely large acceleration or have large inertia, as for
weaponry system or a bridge crane. What’s more, the vibration of the structure induced by a
sequence of subsystems is highly dependent on the system parameters and may develop into
parametric resonance, causing significantly large displacement and stress in the structure.
Therefore, dynamic analysis of such kind of coupled structure-moving subsystem is important to
optimal design and safety consideration.
The coupled structure-moving vehicle system has been demonstrated in detail in previous
chapters. Different from a vehicle-bridge interaction problem, the projection system is modeled as
a cantilever beam where the muzzle or exit end is subject to free boundary condition. For a
cantilever beam, the free end (muzzle) is antinode for all vibration modes, whose vibration has the
greatest effect on the launching accuracy. It is important to study the dynamic response of a gun
barrel or launching structure for a projection system to suppress the muzzle vibration caused by
shooting or launching. By modal impact testing, Littlefield et al. studied the effect of a vibration
absorber on the muzzle brake of an M242 Bushmaster [71]. Application of muzzle-end vibration
absorber was studied by Kathe [72] to constitute a stabilization approach that focused on passive
mechanical structural modification of gun barrel. Tawfik investigated the stability of a stepped gun
barrel with moving bullet using finite element method and eigenvalue approach [73]. Dynamic
124
behavior of electromagnetic railgun was studied by modeling the projectiles as magnetic pressure,
and resonance was observed when armature reached its critical speed [74,75]. With a genetic
algorithm, Esen [76] applied optimization to a passive vibration absorber of a barrel and achieved
a more accurate result compared with experimental studies.
In those studies, a projectile is modeled as a moving mass or moving magnetic pressure (for
railgun). Inertia effect is not considered for magnetic pressure and rigid contact is assumed for the
moving mass model. However, the vibration of the projectile is coupled with the vibration of the
gun barrel through the contact deformation between them during the launching process. This
coupled motion could affect the shooting or launching pattern of projectiles and significantly
influence the accuracy of shooting. What’s more, a repeated launching process could induce
parametric resonance of the gun barrel with ever-increasing amplitude. Therefore, it is necessary
to consider the flexible interaction between a projectile and the gun barrel and study the dynamic
response of the coupled system under repeated launching.
In this chapter, a coupled cantilever beam-accelerating rigid bodies model is developed for
the projection system under repeated launching. In the development, an extended solution domain
(ESD) [39] is established, in which the time-varying coupling between gun barrel and projectiles
can be systematically treated. Dynamic analysis is applied to the coupled system for various
launching rates and accelerations. Parametric resonance is observed in the gun barrel when it is
under 20 repeated launchings. Without proper vibration absorber or damping, the amplitude of
induced vibration at the muzzle is ever-increasing and significantly large. Numerical results on the
parametric excitation show that the occurrence of parametric resonance is highly dependent on
system parameters and cannot be simply predicted by virtual frequency determined in terms of
launching rate.
125
5.2 Problem Statement
A projection system in consideration is shown in Fig. 5.1, where a projectile is accelerated
inside a gun barrel or launching structure. Propulsion of a projectile can be explosion of gunpowder
or electromagnetic thrust. Generally, the gun barrel can fire or launch with an elevation angle
associated with various circumstances. To analyze the dynamic response of barrel and projectiles
as well as the interaction between them, the gun barrel is modeled as a cantilever beam which
undergoes transverse displacement. In this paper, simulation and analysis are carried out for the
projection system to address the issue of parametric resonance under zero elevation angle.
Figure 5.1 Shooting or projection system.
Dynamic response of the gun barrel is studied when it is firing or launching multiple projectiles
in a sequence at a relatively fast rate. For consideration of structure mechanism and system
capability, only one projectile will be launched or fired at one time.
A schematic of the coupled system is shown in Figure 5.2, where the gun barrel is modeled as
a horizontal cantilever Euler-Bernoulli beam with density ( ) x , elastic modulus ( ) E x , cross-
section area ( ) A x and area moment of inertia ( ) I x , and a projectile is modeled as a 2-DOF rigid
body. Interaction between the barrel and projectile is simplified as two point-contacts with
effective stiffness and damping coefficients, regardless of frictions and collisions. Bottom of the
126
gun barrel, where the projectiles are loaded, is assumed to be rigid because its strength is much
larger than the barrel. The ith projectile, moving at a time-dependent speed ( )
i
v t , has vertical
displacement ( )
i
y t at its center of mass, and rotation angle ( )
i
t .
Figure 5.2 Schematic of the coupled system.
Transverse displacement of the cantilever beam is governed by a partial differential equation
(5.1)
2 2 2
2 2 2
( , ) ( , )
( ) ( ) , 0
c
w x t w x t
A x EI x f x L
t x x
(5.1)
where
c
f is resultant force caused by the coupling between rigid bodies and the structure. The
coupling force is in the form
,
2
1 1
,
[ ( ) ( , )]
( ) ( ) ( )
( ) ( , )
ij i G i ij i ij
n
c ij ij
i j
ij i G i ij i ij
k y a a w x t
f x x x t
D
c y a a w x t
Dt
ɺ
ɺ
(5.2)
where ( ) / D Dt is total differentiation, meaning ( , ) / / ( / ) ( / ) Dw x t Dt w t w x dx dt ; ( )
ij
t
are functions of time defined as ( ) ( ) ( )
ij ij ij
t h x h x L , with ( )
ij
x t being the instant location of
jth suspension on the ith rigid body and ( ) h being Heaviside step function; and ( ) is Dirac delta
function.
127
For each rigid body, it undergoes three types of motions depending on its location on the beam
with associated governing equations shown as follows:
Type I (two contact points):
1 i
x L
2
,
1
2
,
1
[ ( ) ( , ) ( )]
( ) ( , ) ( )
i i ij i G i ij i ij ij
j
ij i G i ij i ij ij i
j
m y k y a a w x t t
D
c y a a w x t t m g
Dt
ɺɺ
ɺ
ɺ
(5.3)
2
, , ,
1
2
, ,
1
( )[ ( ) ( , ) ( )]
( ) ( ) ( , ) ( ) 0
c i i ij G i ij i G i ij i ij ij
j
ij G i ij i G i ij i ij ij
j
I k a a y a a w x t t
D
c a a y a a w x t t
Dt
ɺɺ
ɺ
ɺ
(5.4)
Type II (one contact point):
2 1 i i
x L x
2 , 2 2 2
2 , 2 2 2
[ ( ) ( , ) ( )]
( ) ( , ) ( )
i i i i G i i i i i
i i G i i i i i i
m y k y a a w x t t
D
c y a a w x t t m g
Dt
ɺɺ
ɺ
ɺ
(5.5)
, 2 , 2 , 2 2 2
2 , 2 , 2 2 2
( )[ ( ) ( , ) ( )]
( ) ( ) ( , ) ( ) 0
c i i i G i i i G i i i i i
i G i i i G i i i i i
I k a a y a a w x t t
D
c a a y a a w x t t
Dt
ɺɺ
ɺ
ɺ
(5.6)
Type III (no contact points):
2 i
L x
,
, 0
i i i c i i
m y m g I
ɺɺ
ɺɺ (5.7)
For a cantilever beam, fixed-free boundary conditions are specified as follows
2 3
2 3
(0, )
(0, ) 0
( , ) ( , )
0
w t
w t
x
w L t w L t
EI EI
x x
(5.8)
Following assumptions are made regarding initial conditions for the coupled system:
128
• Transverse displacement of the beam is measured from the equilibrium deflection caused
by its weight;
• The beam has zero transverse displacement and velocity at the initial time;
• A rigid body is at equilibrium position due to gravity with zero vertical and angular velocity
when it is loaded before launching.
Hence, the initial conditions of the beam are given as
0
0
( ,0) ( ) 0
( ,0)
( ) 0
w x u x
w x
v x
t
(5.9)
and initial conditions for rigid bodies are as shown in Eqs. (5.10).
2 2
1 , 1 2 , 2
2
1 2 2 1
1 , 1 2 , 2
2
1 2 2 1
( ) ( )
(0) , (0) 0
( )
( ) ( )
(0) , (0) 0
( )
i G i i i G i i
i i i
i i i i
i G i i i G i i
i i i
i i i i
k a a k a a
y m g y
k k a a
k a a k a a
m g
k k a a
ɺ
ɺ
(5.10)
Once a projectile left, the gun barrel has free vibration until the next projectile is loaded and
launched. For a repeated launching process, the gun barrel is under a forced-free-forced vibration
cycle. To determine the dynamic response of the coupled system, it requires to solve a mixed set
of a partial differential equation (5.1) and ordinary differential equations (5.3-5.7) simultaneously.
However, because of the repeated launching process, the beam becomes coupled and decoupled
with each projectile sequentially, which makes an analytical solution for the coupled system
extremely difficult to obtain, if not impossible.
5.3 Solution Method
To analyze the dynamic response of the gun barrel under repeated shooting or launching process,
a cantilever beam coupled with rigid bodies launched in a sequence need to be considered. Three
key issues need to be addressed in the determination of dynamic response for the coupled system.
129
First, the coupled system is governed by a partial differential equation (beam) and a set of ordinary
differential equations (rigid bodies). Those equations are coupled and require being solved
simultaneously. Second, the coupling force between the beam and a rigid body is time-dependent
that makes the system a non-autonomous or time-variant system where general analytical solution
cannot be obtained. Third, the coupling between a rigid body and the beam is very complicated
when it enters the beam from the loading area, or when it leaves the beam into open field. This
makes the governing equations for the coupled system inconsistent during the launching process
and requires an adaption in the simulation algorithm for conventional methods.
The semi-analytical method presented in this paper is developed to resolve those issues. The
solution procedure includes three parts: (a) establishment of an extended solution domain (ESD);
(b) determination of analytical eigensolutions of the beam with distributed transfer function
method (DTFM); (c) formulation of the governing equations for the coupled system with the
generalized assumed-mode method, which are demonstrated in the sequel.
5.3.1 Extended solution domain
The moving rigid body problem on a cantilever beam for projection system represents a more
complicated situation compared with bridge-vehicle interaction problem. The coupling between a
rigid body and the beam has five different stages: (i) the rigid body is in the loading area with no
contact points on the beam; (ii) only the front contact point is on the beam; (iii) two contact points
are both on the beam; (iv) only the rear contact point is on the beam; (v) the rigid body is in the
open field with no contact points on the beam. However, because the cantilever beam has open
right end, a rigid body loses one supporting force once the front contact point passes the right end.
It is very different from a bridge-vehicle interaction problem where the supporting forces between
the rigid body and a rigid surface preserve. Vanishing of supporting forces renders an inconsistent
130
governing equation for a rigid body as shown in Eqs. (5.3-5.7). Additionally, in our problem,
projectiles are firing or launching repeatedly at a certain rate. For a conventional method, it
requires to check the locations of contact points and adjust the governing equations for the coupled
system at every computational step, which is less efficient and systematic.
Figure 5.3 Schematic of extended solution domain for the projection system.
In this section, an altered extended solution domain (ESD) is defined based on the previous
chapter’s work to deal with this complicated coupling between a cantilever beam and repeatedly
launched projectiles. A rigid loading domain
L
where projectiles are loaded or prepared is
defined to the left of the beam domain
B
. Within the rigid loading domain
L
, a projectile is at
equilibrium with zero motion until it is pushed into beam domain (gun barrel) by explosion or
magnetic thrust. As shown in Figure 5.3, when the launching process starts, the front contact point
of a corresponding projectile is at the left end of the beam domain (x = 0) indicating flexible
coupling occurs between the projectile and the beam. A virtual rigid domain
R
is defined to the
right of the beam domain where a projectile is connected by a virtual spring with zero stiffness
coefficient. When the front contact point of a projectile moves into the virtual domain
R
, it loses
one contact with the beam which is interpreted by Eqs. (5.5) and (5.6). The rigid body loses both
contacts with the beam once the rear contact point moves into the virtual domain
R
which is
131
described by Eq. (5.7). The virtual domain
R
is separated from the right end of the beam so that
the free-end boundary conditions are not violated.
The extended solution domain (ESD) is a union of three subdomains as given below
Beam structure domain: { | 0 }
Left extended domain: { | 0}
Right extended domain: { | }
B
L L
R R
x x L
x D x
x L x L D
(5.11)
where the length of the extended domain is determined as
12 11 1
0
, ( )
f
t
L R
D a a D v t dt L
(5.12)
with
f
t being the total time for the launching process.
With the ESD so determined, an extended structure displacement ( , ) W x t is defined for the
entire extended solution domain as
( , )
( , )
0
B
L R
x w x t
W x t
x
(5.13)
where ( , ) w x t is transverse displacement of the beam governed by Eq. (5.1). It is obvious to see
that all rigid bodies (projectiles) are coupled with the extended structure within the extended
solution domain during the entire repeated launching process. With this modeling technique, a
consistent coupled model can be established for the projection system with a constant number of
coupled projectiles and a consistent number of degrees of freedom. It provides a convenient way
to analyze the dynamic response for the projection system and the interaction between projectiles
and the supporting structure when it undergoes a repeated launching process.
5.3.2 Analytical eigensolutions of a cantilever beam
The eigenvalue problem for a uniform cantilever beam is defined as
132
4
2
4
( )
( ) 0, 0
d x
A x EI x L
dx
(5.14)
with boundary conditions as
2 3
2 3
(0)
(0) 0
( ) (0)
0
d
dx
d L d
dx dx
(5.15)
where is an eigenvalue (natural frequency) and ( ) x is the associated eigenfunction (mode
shape); , , , A E I are the beam parameters defined in Section 5.2. Using the formulation of DTFM
[52,55,56], the eigenvalue problem is written in the state form
( ) ( ) ( ), 0
d
x F x x L
dx
(5.16)
where ( ) x is a state vector defined as
2 3
2 3
( )
w w w
x w
x x x
(5.17)
and ( ) F is a four-by-four matrix constituted of system parameters and natural frequency . The
boundary conditions are written in matrix form as
(0) ( ) 0 M N L (5.18)
where M and N are boundary matrices
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
,
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
M N
(5.19)
The solution to Eq. (5.16) gives the analytical eigensolution of the cantilever beam which is in
the form ( ) ( ) x x where ( ) x is a four-by-four matrix as below with being
nondimensional eigenvalue and is a coefficient vector to be determined.
133
2 2 2 2
3 3 3 3
cos( ) sin( )
sin( ) cos( )
( )
cos( ) sin( )
sin( ) cos( )
x x
x x
x x
x x
x x e e
x x e e
x
x x e e
x x e e
(5.20)
Substituting the analytical eigensolution into boundary conditions (5.18) yields the
characteristic equation (5.21) which is a transcendental equation and whose solution gives the
natural frequencies.
( ) det (0) ( ) 0 M N L (5.21)
The coefficient vector is solved from a homogeneous equation (5.22).
(0) ( ) 0 M N L (5.22)
With so determined, eigenfunction or mode shape for the cantilever beam is directly
obtained from the analytical eigensolution ( ) x as
( ) 1 0 0 0 ( ) x x (5.23)
Once the analytical eigenfunctions for the beam structure obtained, a discretized governing
equation is derived with the generalized assumed-mode method, whose solution gives the dynamic
response for the coupled system. The detailed derivation shall be seen in the sequel.
5.3.3 Generalized assumed-mode method
As mentioned in the previous section, the coupled system is governed by a partial differential
equation (5.1) and a set of ordinary differential equations (5.3-5.7) which are coupled by time-
variant functions. It is extremely difficult to obtain a general analytical solution for such a non-
autonomous system if not impossible. Hence, a semi-analytical method is proposed to construct a
discretized governing equation via the generalized assumed-mode method.
134
Different from a conventional assumed-mode method where admissible functions are used, the
generalized assumed-mode method uses analytical eigenfunctions (mode shapes) of the beam
which are comparison functions that satisfy both essential boundary conditions and natural
boundary conditions. It can achieve higher accuracy and efficiency in numerical simulation. The
analytical eigensolution for a cantilever beam is derived using the distributed transfer function
method as discussed in Section 5.3.2.
With the ESD, extended eigenfunctions are defined in Eq. (5.24)
( , )
( , )
0
B
L R
x x t
x t
x
(5.24)
where ( ) x is the eigenfunction of the cantilever beam. To formulate the discretized governing
equation, a truncated series expression for the extended structure displacement is defined with the
generalized assumed-mode method as
( ) ( )
1
( , ) ( ) ( ) ( ) ( ),
m
k k
k
W x t x q t x q t x
(5.25)
where
( )
( )
k
x is the extended eigenfunction (mode shape) associated with the kth mode and
( )
( )
k
q x is the associated modal displacement; ( ) x and ( ) q x are vectors of the form
(1) (2) ( )
(1) (2) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
m
T
m
x x x x
q t q t q t q t
⋯
⋯
(5.26)
The discretized governing equation is derived in Eq. (5.27) by substituting the truncated series
(5.25) into energy functionals and applying extended Hamilton’s principle
1
1 2
0 0 ( ) ( ) ( )
0 ( ) ( ) ( )
T T
b s c b s c
g r c r c c r
M q t q t q t C C K K K
f M s t s t s t C C K K K
ɺɺ ɺ
ɺɺ ɺ
(5.27)
135
where s is a vector of the displacements and rotation angles of the rigid bodies and
g
f is a vector
of external forces, which are given by
1 1 2 2
1 2
0 0 0
T
n n
T
g n
s y y y
f m g m g m g
⋯
⋯
(5.28)
In Eq. (5.27),
r
M ,
r
C and
r
K are inertia, damping and stiffness matrices of moving rigid
bodies, given by
1 ,1 ,
1 1 ,1 1
2
1 ,1 1 1 ,1 1
2
1
,
2
, ,
1 1 ,1 1
2
1 ,1 1 1 ,1 1
,
diag
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )
r c n c n
j j G j
j G j j G j
r
j
nj nj G n nj
nj G n nj nj G n nj
j j G j
j G j j G j
r
nj nj G n nj
n
M m I m I
c c a a
c a a c a a
C
c c a a
c a a c a a
k k a a
k a a k a a
K
k k a a
k
⋯
⋱
⋱
2
1
2
, ,
( ) ( )
j
j G n nj nj G n nj
a a k a a
(5.29)
and
1 2
, , , ,
s c s c c
C C K K K are coupling damping and stiffness matrices due to beam-rigid body
interactions, given by;
2 2
1 1 1 1
1 1
2 2
1 ,1 1 1 1 ,1 1 1
1 1
1
2 2
1 1
2 2
, ,
1 1
( ) ( )
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( )
j j j j
j j
j G j j j G j j
j j
c c
nj nj nj nj
j j
nj G n nj nj nj G n nj nj
j j
c x k x
c a a x k a a x
C K
c x k x
c a a x k a a x
⋮ ⋮
136
2 2
1 1 1 1
( ) ( ), ( ) ( ) ( ) '( )
n n
T T T
s ij ij ij s ij ij ij i ij ij ij
i j i j
C c x x K k x x v c x x
2
1 1 1
1
2
1 1 ,1 1 1
1
2
2
1
2
,
1
'( )
( ) '( )
'( )
( ) '( )
j j
j
j G j j
j
c
nj n nj
j
nj n G n nj nj
j
c v x
c v a a x
K
c v x
c v a a x
⋮ (5.30)
The initial condition of Eq. (5.27) is
(0) (0) (0) (0) (0)
T
T T T T
z q s q s
ɺ ɺ (5.31)
with
0 0
(0) ( ) ( ) ( ) , (0) ( ) ( ) ( )
T T
q A x u x x dx q A x v x x dx
ɺ (5.32)
Solving Eq. (5.27) subject to initial conditions (5.10), (5.31) and (5.32) via numerical
simulation gives the dynamic response for the coupled system. In summary, solving the dynamic
response for the projection system follows four steps:
(i). Determine the analytical eigenfunctions (mode shapes) for the cantilever beam;
(ii). Define the extended solution domain (ESD);
(iii). Establish the discretized governing equation by the generalized assumed-mode method;
(iv). Solve the dynamic response via numerical simulation.
With the proposed semi-analytical method, dynamic response for a projection system can be
determined efficiently and accurately with no difficulties dealing with coupling between multiple
contact points and the gun barrel. Numerical results are presented and discussed in the next section.
137
5.4 Numerical Results
In this section, the proposed method is illustrated through investigation of the transient response
of a gun barrel under repeated launching process. A gun barrel is modeled as a cantilever beam
with modal damping whose parameters are defined as follows:
3 3
2 5 4
7.83 10 kg/m , 200 Gpa
0.02 m , 2 10 m , 6 m
E
A I L
The projectile is assumed to be uniformly accelerating inside the barrel and moving with a constant
speed once launched from the left end of the beam. The launching rate is determined by the time
between two launchings denoted by
l
t .
Numerical simulation is carried out for two cases. First, a demonstration case is considered for
various accelerations and launching rates, showing the possible influences that projectiles could
have on the gun barrel vibration. Second, parametric excitation is studied when the gun barrel is
under repeated launching condition. Vibration with ever-increasing amplitude and parametric
resonance is observed in the launching structure.
5.4.1 Effect of acceleration and launching rate
In this example, the effect of acceleration and launching rate is studied. First, consider a single
projectile case for four different accelerations, 10000,20000,30000 a and
2
50000 m/s . The
projectile parameters are listed in Table 5.1 where m is mass,
c
I is moment of inertia about its
center of mass, D is length,
G
a is the distance from the projectile head to the center of mass,
j
k
and
j
c are spring coefficient and damping coefficient of the jth contact point and
j
a is the distance
from the head to the jth contact point.
Table 5.1 Parameters of projectile.
Parameters Value
(kg) 30
138
(kg ∙ m
) 3
(cm) 50
() 25
,
(N/m) 1 × 10
,
(N ∙ s/m) 100
(cm) 10
(cm) 40
Figure 5.4 Displacement of beam tip for various projectile accelerations.
Vertical displacement of the beam tip is plotted versus the normalized time /
f
t t for four
accelerations in Figure 5.4, where
f
t is the final time when the rear contact point of projectile
reaches the right end of the beam. A smaller acceleration (
2
20000 m/s a ) associated with a
slower muzzle velocity ( 346.41 m/s
m
v ) induces a larger deflection at beam tip as shown in
Figure 5.4. However, for a larger acceleration (
2
50000 m/s a ) associated with a faster muzzle
velocity ( 774.60 m/s
m
v ), the downward displacement of the beam tip is larger than other cases
even its maximum displacement is smaller.
139
(a)
(b)
140
(c)
Figure 5.5 Displacement of beam tip induced by five projectiles with various launching rate:
(a) 2 s
l
t ; (b) 3 s
l
t ; (c) 5 s
l
t .
Dynamic response of the beam induced by a sequence of projectiles is very different from that
induced by only one projectile. In the following example, five projectiles are launched sequentially
with various launching rate
l
t . Vertical displacement of the beam tip is plotted for three cases,
2,3
l
t and 5 s in Figure 5.5 (a), (b) and (c) respectively.
From the figure, it is obvious that the beam vibration induced by a sequence of projectiles is
very different under various launching rate. Displacement of the beam tip shows an ever-increasing
amplitude when the projectile is launched every 2 s; seen in Figure 5.5(a). Displacement of the tip
has a very similar pattern with insignificant deviation in amplitude for launching rate being 3 s and
5 s as shown in Figure 5.5 (b) and (c). When the gun barrel undertakes repeatedly launched
projectiles, vibration induced by those projectiles highly depends on the system parameters and
141
launching pattern. Under certain circumstance, the beam could have vibration with ever-increasing
amplitude even resonance. For a coupled beam-projectile system, this kind of resonance is
different from the case when the beam is subject to an external excitation whose frequency
coincides with one of the beam’s natural frequencies. Identified as parametric resonance, a similar
phenomenon has been observed in a coupled structure-moving vehicle system in chapter 4.
5.4.2 Parametric resonance projectile induced vibration
For a projection system, repeatedly launched projectiles could induce vibration of the beam
with ever-increasing amplitude, leading to parametric resonance. However, unlike the moving
force problem where the moving force can be decomposed as periodic external forces, a projection
system where projectiles are modeled as moving rigid bodies cannot be solved analytically. In this
section, some parametric studies are carried out to demonstrate the relation between system
parameters and the parametric resonance. In the simulation, 20 projectiles are launched from the
gun barrel in a sequence with constant acceleration
2
20000 m/s a at various launching rates. The
parameters of the gun barrel are the same as the previous example but with zero modal damping
and parameters of the projectile are the same as listed in Table 5.1.
Vertical displacements of the beam tip induced by 20 projectiles with launching rate being
2,3.2
l
t and 3.5 s are plotted in Figures. 5.6, 5.7 and 5.8 respectively. The amplitude of the
induced vibration at the beam tip is ever-increasing for 2 s
l
t ; seen in Figure 5.6. For launching
rate 3.2 s
l
t , the vibration of the beam tip is increasing at a slower rate and eventually becomes
relatively stable with an insignificant change in amplitude. For launching rate 3.5 s
l
t , the
vibration of the beam tip is bounded and less than
3
1 10
m for the entire launching process.
142
Figure 5.6 Beam tip displacement (zero modal damping) induced by 20 projectiles with 2 s
l
t .
Figure 5.7 Beam tip displacement (zero modal damping) induced by 20 projectiles with 3.2 s
l
t .
Figure 5.8 Beam tip displacement (zero modal damping) induced by 20 projectiles with 3.5 s
l
t .
143
Figure 5.9 Beam tip displacement (1% modal damping) induced by 20 projectiles with 2 s
l
t .
Figure 5.10 Beam tip displacement (1% modal damping) induced by 20 projectiles with 3.2 s
l
t .
Figure 5.11 Beam tip displacement (1% modal damping) induced by 20 projectiles with 3.5 s
l
t .
144
By adding 1% modal damping to the structure, the dynamic performance of the cantilever beam
is improved as shown in Figures. 5.9, 5.10 and 5.11. In Figure 5.9, displacement of the beam tip
is still increasing gradually when launching rate is 2s. But the maximum amplitude of the tip
displacement is reduced significantly. As for launching rate being 3.2s and 3.5s, patterns of tip
displacement are altered as well, seen in Figures. 5.10 and 5.11, compared with those from a
structure with zero modal damping.
Table 5.2 First eight natural frequencies of the cantilever beam.
Mode
Natural frequency
(rad/s)
1 15.609
2 97.822
3 273.90
4 536.74
5 887.27
6 1325.4
7 1851.2
8 2464.6
The first eight natural frequencies of the cantilever beam are listed in Table 5.2. Virtual
frequency is defined in terms of the launching rate 2 /
r l
t to describe the launching pattern.
For the case when parametric resonance occurs (seen in Figure 5.6), virtual frequency is
3.1416 rad/s
r
, which is much less than the first natural frequency of the beam. Since the
coupled system is a time-variant (nonautonomous) system, its parametric resonance cannot be
simply predicted by the resonance condition obtained with a moving force model.
Parametric resonance is highly dependent on the system parameters, including launching rate,
acceleration of projectiles, length of the gun barrel, etc. It is very different from the conventional
resonance where the excitation frequency matches one of the system’s natural frequencies. Since
the coupled system in consideration is time-variant, it is very difficult to obtain a general analytical
solution for dynamic response. By numerically simulating the dynamic response of gun barrel
145
under repeated launchings, qualitative relations between parametric resonance and system
parameters can be found.
5.5 Summary
A semi-analytical method is developed for modeling and dynamic analysis of a projection
system subject repeatedly launched projectiles. By modeling each projectile as a rigid body with
two contact points on the gun barrel, coupled motion for gun barrel and projectiles are considered
simultaneously. Since the coupling between a projectile and the supporting beam is quite
complicated due to the changing number of contact points, a conventional method needs to check
the stage of the coupling and apply adaption to the mathematical model for the coupled system.
The proposed method resolves this issue by defining an extended solution domain (ESD), which
implements a systematic description of beam-projectile interaction. With the ESD-based modeling
and solution technique, dynamic response of the coupled system is solved by applying the
distributed transfer function method (DTFM) and the generalized assumed-mode method.
Numerical simulation is carried out for a demo example with five projectiles under various
launching rates and accelerations. Results show that the beam can have induced vibration with
ever-increasing amplitude which has the potential to become resonance. Based on those results, an
extended case with 20 projectiles is considered for parametric excitation study. According to the
numerical results, parametric resonance which has ever-increasing and significantly large
amplitude can be induced in the beam structure by repeatedly launched projectiles. The parametric
resonance is different from conventional resonance where the excitation frequency coincides with
one of the system’s natural frequencies. The virtual frequency determined by the launching rate is
much less than the first natural frequency of the beam.
146
Chapter 6 Conclusion
In this study, a supporting structure being considered for a coupled structure-moving subsystem
problem is modeled as a multi-span stepped beam with elastic column supports. This kind of
modeling represents very commonly seen applications, such as elevated bridges or railways.
Lumped model is applied for moving subsystem, with considering the inertia effect of the
subsystem and flexible structure-subsystem interactions. To describe the flexible structure-
subsystem interaction, 1 DOF oscillator model and 2 DOF rigid body model are considered.
Dynamic response of the supporting structure induced by moving subsystem is the interests of this
work.
Dynamic response of the supporting structure induced by a sequence of moving subsystems is
very different from that induced by single or just a few moving subsystems. However, due to the
complex coupling situation caused by general passing pattern, the number of total degrees of
freedom for the coupled system changes frequently. This is especially true for moving rigid body
model because multiple contact points exist between the structure and the rigid body. To get a
consistent and systematic formulation for the structure-subsystem interaction, an extended solution
domain (ESD) technique is developed. By defining two virtual domains which are assumed to be
rigid surfaces, a union of three subdomains, the beam domain
B
and two virtual domains ,
L R
,
is generated for the coupled system. The virtual domains are chosen to be the minimum range that
could “park” all moving subsystems at the initial time (t = 0) and final time (
f
t t ). With the ESD
so determined, all moving subsystem will be coupled with an extended structure which is defined
within the ESD. This formulation guarantees a fixed number of degrees of freedom and a constant
dimension of matrices. With the help of ESD, a systematic mathematical model can be built to
147
describe the complex structure-subsystem coupling and it does not require checking the total
number of degrees of freedom for the coupled system nor adaption of the solution algorithm.
The system in consideration is a coupled distributed-lumped system, which is governed by a
partial differential equation (beam structure) and a set of ordinary differential equations
(subsystems). It requires solving this mixed set of differential equations simultaneously. What’s
more, due to the time-varying locations of moving subsystems, the coupled system is a
nonautonomous (time-variant) system, whose general analytical solution is not accessible. To
solve the dynamic response of the coupled time-variant system, a semi-analytical solution method
is proposed.
This method is a combination of the distributed transfer function method (DTFM), which is
analytical and closed form, and the generalized assumed-mode method, which is approximate and
numerical. The DTFM is used to get the analytical eigenfunctions (mode shapes) for the multi-
span beam structure with column supports. With an augmented formulation of DTFM, analytical
eigenfunctions are derived without using matrix exponentials which may result in numerical
instability and errors. Eigenfunctions formulated with DTFM can be applied to arbitrary boundary
conditions, which are not discussed in this work. Once analytical eigenfunctions obtained through
DTFM, a discretized formulation for the coupled system is derived using the generalized assumed-
mode method. Different from the conventional assumed-mode method where admissible functions
are used, the generalized assumed-mode method utilizes the analytical eigenfunctions for the beam
which are comparison functions that satisfy both essential boundary conditions and natural
boundary conditions. By applying comparison functions, the generalized assumed-mode method
can achieve better efficiency and accuracy in the dynamic simulation. Dynamic response of the
148
coupled system is obtained by solving the discretized governing equations via numerical methods.
In this work, the fourth-order Runge-Kutta method is used.
Convergence study of the proposed method is carried out for moving oscillator model. Because
the analytical eigenfunctions are used in the generalized assumed-mode formulation, convergence
of the proposed method is fast, indicating that a relatively small number of modes are needed in
this formulation to get sufficiently accurate results. Numerical results obtained with the proposed
method are compared with those from the finite element method (FEM). The comparison shows
that the proposed method is much more efficient than conventional FEM, being capable to save at
least 80% computational time if the same accuracy is guaranteed.
Numerical results of some case studies show that a sequence of moving subsystems (oscillator
or rigid body model) could induce structure vibration with ever-increasing and significantly large
amplitude. Parametric analysis on this kind of vehicle-induced resonance is done with moving
rigid body model. By investigating the maximum displacement of the structure induced by a
sequence of rigid bodies moving at various speeds, resonance has been detected from the transient
response history. This resonance, named as parametric resonance, is caused by repeated changing
of system configuration and different from conventional resonance caused by external excitation
at a certain frequency. It has been shown from the parametric analysis that the vehicle-induced
resonance is highly dependent on system parameters. Because the coupled system is time-variant,
no general analytical relation between the parametric resonance and system parameters has been
discovered.
To get a better prediction of parametric resonance caused by the repeated passage of moving
subsystems, a semi-analytical method is built using the moving oscillator model. The proposed
method is based on the semi-analytical solution method developed for the structure-moving
149
subsystem problem. By using mapping transformation formulation, an analytical resonance
criterion is derived which only requires evaluation of eigenvalues of the mapping matrix. The
structure is going to have parametric resonance if the spectral radius of the mapping matrix is
greater than one. The proposed method is semi-analytical because the calculation of the mapping
matrix is numerical or approximate since the transition matrix has no general analytical
expressions for a time-variant system. Even though, the proposed method provides a much more
efficient approach for parametric resonance prediction because it does not require dynamic
simulation for the complete passing process but only for one period of the repeated passage. This
semi-analytical method is capable to not only predict the parametric resonance but also evaluate
the steady state value of structure response if it is stable and bounded. With mapping matrix Z
obtained from one-period simulation, the steady state response can be predicted by
1
(I ) Z
.
Although, the proposed semi-analytical method is derived from the moving oscillator model. It
can be extended to moving rigid body model, and many other extended cases. What’s more, for a
fast projection system discussed in Chapter 5, parametric resonance induced by shooting
projectiles can also be predicted by the resonance criterion derived with the mapping
transformation. More research work is still needed for further investigation.
150
References
1. G. Diana, F. Cheli, Dynamic interaction of railway systems with large bridges, Vehicle System
Dynamics 18 (1-3) (1989) 71-106.
2. Y. Cai, S.S. Chen, D.M. Rote, H.T. Coffey, Vehicle/guideway interaction for high speed
vehicles on a flexible guideway, Journal of Sound and Vibration 175 (1994) 625-646.
3. W.D. Zhu, C.D. Mote, Free and forced response of an axially moving string transporting a
damped linear oscillator, Journal of Sound and Vibration 177 (1994) 591-610.
4. Q.C. Nguyen, K.S. Hong, Simultaneous control of longitudinal and transverse vibrations of an
axially moving string with velocity tracking, Journal of Sound and Vibration 331 (2012) 3006-
3019.
5. D. Hochlenert, G. Spelsberg-Korspeter, P. Hagedorn, Friction induced vibrations in moving
continua and their application to brake squeal, Journal of Applied Mechanics 74 (2007) 542.
6. C. R. Steele, The finite beam with a moving load, Journal of Applied Mechanics 34 (1967)
111-118.
7. J.D. Achenbach, C.T. Sun, Moving load on a flexibly supported Timoshenko beam,
International Journal of Solids and Structures 1 (1965) 353-370.
8. L. Fryba, Vibration of Solids and Structures under Moving Loads (Thomas Telford, London,
1973).
9. K. Henchi, M. Fafard, G. Dhatt, M. Talbot, Dynamic behavior of multi-span beams under
moving loads, Journal of Sound and Vibration 199 (1997) 33-50.
10. D. Y. Zheng, Y. K. Cheung, F. T. K. Au, Y. S. Cheng, Vibration of multi-span non-uniform
beams under moving loads by using modified beam vibration functions, Journal of Sound and
Vibration 212 (1998) 455-467.
151
11. Y. Song, T. Kim, U. Lee, Vibration of a beam subjected to a moving force: Frequency-domain
spectral element modeling and analysis, International Journal of Mechanical Sciences 113
(2016) 162-174.
12. P. Castro Jorge, F. M. F. Simoes, A. Pinto da Costa, Dynamics of beams on non-uniform
nonlinear foundations subjected moving loads, Computers and Structures 148 (2015) 26-34.
13. P. Museros, E. Moliner, M. D. Martinez-Rodrigo, Free vibration of simply-supported beam
bridges under moving loads: maximum resonance, cancellation and resonant vertical
acceleration, Journal of Sound and Vibration 332 (2013) 326-345.
14. Y. B. Yang, J. D. Yau, Resonance of high-speed trains moving over a series of simple or
continuous beams with non-ballasted tracks, Engineering Structures 143 (2017) 295-305.
15. E.C. Ting, J. Genin, J.H. Ginsberg, A general algorithm for moving mass problems, Journal of
Sound and Vibration 33 (1974) 49-58.
16. U. Lee, Separation between the flexible structure and the moving mass sliding on it, Journal
of Sound and Vibration 209 (1998) 867-877.
17. T.E. Blejwas, R.S. Ayre, Dynamic interaction of moving vehicles and structures, Journal of
Sound and Vibration 67 (1979) 513-521.
18. J.E. Akin, M. Mofid, Numerical solution for response of beams with moving mass, Journal of
Structural Engineering 115 (1989) 120-131.
19. A. V Pesterev, L.A. Bergman, A contribution to the moving mass problem, Journal of
Vibration and Acoustics 120 (1998) 824-826.
20. M. Ichikama, Y. Miyakawa, A. Matsuda, Vibration analysis of the continuous beam subjected
to a moving mass, Journal of Sound and Vibration 230 (2000) 493-506.
152
21. A. Nikkhoo, F.R. Rofooei, M.R. Shadnam, Dynamic behavior and modal control of beams
under moving mass, Journal of Sound and Vibration 306 (2007) 712-724.
22. M. Mofid, J.E. Akin, Discrete element response of beams with traveling mass, Advances in
Engineering Software 25 (2-3) (1996) 321-331.
23. A. Yavari, M. Nouri, M. Mofid, Discrete element analysis of dynamic response of Timoshenko
beams under moving mass, Advances in Engineering Software 33 (2002) 143-153.
24. J.V. Amiri, A. Nikkhoo, M.R. Davoodi, M.E. Hassanabadi, Vibration analysis of a Mindlin
elastic plate under a moving mass excitation by eigenfunction expansion method, Thin-Walled
Structures 62 (2013) 53-64.
25. A. H. Karimi, S. Ziaei-Rad, Vibration analysis of a beam with moving support subjected to a
moving mass travelling with constant and variable speed, Communications in Nonlinear
Science and Numerical Simulation 29 (2015) 372-390.
26. Z. Dimitrovova, New semi-analytical solution for a uniformly moving mass on a beam on a
two-parameter visco-elastic foundation, International Journal of Mechanical Sciences 127
(2017) 142-162.
27. M. Klasztorny, J. Langer, Dynamic response of single-span beam bridges to a series of moving
loads, Earthquake Engineering & Structural Dynamics 19 (1990) 1107-1124.
28. Y.-H. Lin, M.W. Trethewey, Finite element analysis of elastic beams subjected to moving
dynamic loads, Journal of Sound and Vibration 136 (1990) 323-342.
29. Y. B. Yang, B. H. Lin, Vehicle-bridge interaction analysis by dynamic condensation method,
Journal of Structural Engineering 121 (1995) 1636-1643.
30. Y. B. Yang, J. D. Yau, Vehicle-bridge interaction element for dynamic analysis, Journal of
Structural Engineering 123 (1997) 1512-1518.
153
31. A. V. Pesterev, L. A. Bergman, Response of elastic continuum carrying moving linear
oscillator, Journal of Engineering Mechanics 123 (1997) 878-884.
32. A. V. Pesterev, L. A. Bergman, Vibration of elastic continuum carrying accelerating oscillator,
Journal of Engineering Mechanics 123 (1997) 886-889.
33. B. Yang, C.A. Tan, L.A. Bergman, Direct numerical procedure for solution of moving
oscillator problems, Journal of Engineering Mechanics 126 (2000) 462-469.
34. L.A. Bergman, B. Yang, C.A. Tan, Response of elastic continuum carrying multiple moving
oscillators, Journal of Engineering Mechanics 127 (2001) 260-265.
35. M. F. Green, D. Cebon, Dynamic interaction between heavy vehicles and highway bridges,
Computers and Structures 62 (1997) 253-264.
36. K. Rajabi, M. H. Kargarnovin, M. Gharini, Dynamic analysis of a functionally graded simply-
supported euler-bernoulli beam subject to a moving oscillator, Acta Mechanica 224 (2013)
425-446.
37. N.D. Zrnic, V.M. Gasic, S.M. Bosnjak, Dynamic response of a gantry crane system due to a
moving body considered as moving oscillator, Archive of Civil and Mechanical Engineering
15 (2015) 243-250.
38. Y. Wu, Y. Gao, Dynamic response of a simply supported viscous damped double-beam system
under the moving oscillator, Journal of Sound and Vibration 384 (2016) 194-209.
39. B. Yang, H. Gao, S. Liu, Vibrations of a multi-span beam structure carrying many moving
oscillators, International Journal of Structural Stability and Dynamics 18 (2018) 1850125.
40. Y. Chen, C.A. Tan, L.A. Bergman, Effects of boundary flexibility on the vibration of a
continuum with a moving oscillator, Journal of Vibration and Acoustics 124 (2002) 552-560.
154
41. G. Muscolino, S. Benfratello, A. Sidoti, Dynamics analysis of distributed parameter system
subjected to a moving oscillator with random mass, velocity and acceleration, Probabilistic
Engineering Mechanics 17 (2001) 63-72.
42. D. Stǎncioiu, H. Ouyang, J.E. Mottershead, Vibration of a beam excited by a moving oscillator
considering separation and reattachment, Journal of Sound and Vibration 310 (2008) 1128-
1140.
43. S. S. Law, X. Q. Zhu, Bridge Dynamic response due to road surface roughness and braking of
vehicle, Journal of Sound and Vibration 282 (2005) 805-830.
44. P. Lou, Finite Element Analysis for train-track-bridge interaction system, Archive of Applied
Mechanics 77 (2007) 707-728.
45. P. Lou, F. T. K. Au, Finite element formulae for internal forces of Bernoulli-Euler beams under
moving vehicles, Journal of Sound and Vibration 332 (2013) 1533-1552.
46. K. Liu, G. De Roeck, G. Lombaert, The effect of dynamic train-bridge interaction on the bridge
response during a train passage, Journal of Sound and Vibration 325 (2009) 240-251.
47. Y. Chen, B. Zhang, N. Zhang, M. Zheng, A condensation method for the dynamic analysis of
vertical vehicle-track interaction considering vehicle flexibility, Journal of Vibration and
Acoustics137 (2015) 041010.
48. M. Fedorova, M. V. Sivaselvan, An algorithm for dynamic vehicle-track-structure interaction
analysis for high-speed trains. Engineering Structures 148 (2017) 857-877.
49. Q. Zeng, C. D. Stoura, E. G. Dimitrakopoulos, A localized Lagrange multipliers approach for
the problem of vehicle-bridge-interaction. Engineering Structures 168 (2018) 82-92.
155
50. J. M. Olmos, M. Astiz, Non-linear vehicle-bridge-wind interaction model for running safety
assessment of high-speed trains over a high-pier viaduct. Journal of Sound and Vibration 419
(2018) 63-89.
51. W. Wang, Y. Zhang, H. Ouyang, An iterative method for solving the dynamic response of
railway vehicle-track coupled systems based on prediction of wheel-rail forces. Engineering
Structures 151 (2017) 297-311.
52. B. Yang, C, A. Tan, Transfer functions of one-dimensional distributed parameter systems,
Journal of Applied Mechanics 59 (1992) 1009-1014.
53. E.J. Hearn, Mechanics of Materials, Vol. 1 (Butterworth-Heinemann, UK, 1985).
54. S.S. Rao, Vibration of Continuous Systems (John Wiley & Sons, 2007).
55. B. Yang, Stress, strain, and structural dynamics: an interactive handbook of formulas, solutions
and MATLAB toolboxes (Elsevier Academic Press, Oxford, UK, 2005).
56. B. Yang, Distributed transfer function analysis of complex distributed parameter systems,
Journal of Applied Mechanics 61 (1994) 84-92.
57. R. D. Cook, Concepts and Applications of Finite Element Analysis (John Wiley & Sons, 2007).
58. Y. B. Yang, C. W. Lin, Vehicle-bridge interaction dynamics and potential applications, Journal
of Sound and Vibration 284 (2005) 205-226.
59. K. Noh and B. Yang, An augmented state formulation for modeling and analysis of multibody
distributed dynamic systems, Journal of Applied Mechanics 81 (2014) 051011.
60. Y. B. Yang, J. D. Yau and L. C. Hsu, Vibration of simple beams due to trains moving at high
speeds, Engineering Structures 19 (1997) 936-944.
61. J. D. Yau, Y. S. Wu and Y. B. Yang, Impact response of bridges with elastic bearings to moving
loads, Journal of Sound and Vibration 248 (2001) 9-30.
156
62. Y. B. Yang, C. L. Lin, J. D. Yau and D. W. Chang, Mechanism of resonance and cancellation
for train-induced vibration on bridges with elastic bearings, Journal of Sound and Vibration
269 (2004) 345-360.
63. L. Mao and Y. Lu, Critical speed and resonance criteria of railway bridge response to moving
trains, Journal of Bridge Engineering 18 (2013) 131-141.
64. M. Pirmoradian, M. Keshmiri and H. Karimpour, On the parametric excitation of a
Timoshenko beam due to intermittent passage of moving masses: instability and resonance
analysis, Acta Mechanica 226 (2015) 1241-1253.
65. A. Nikkhoo and F. Rofooei, Parametric study of the dynamic response of thin rectangular
plates traversed by a moving mass, Acta Mechanica 223 (2012) 15-27.
66. M. E. Hassanabadi, N. K. A. Attari, A. Nikkhoo and S. Mariani, Resonance of a rectangular
plate influenced by sequential moving masses, Coupled Systems Mechanics 5 (2016) 87-100.
67. E. Torkan, M. Pirmoradian and M. Hashemian, On the parametric and external resonances of
rectangular plates on an elastic foundation traversed by sequential masses, Archive of Applied
Mechanics 88 (2018) 1411-1428.
68. J. D. Yau and Y. B. Yang, Resonance of a series of train cars traveling over multi-span
continuous beams, Dynamics and Control of Advanced Structures and Machines (2017) 11-
21.
69. Y. B. Yang and J. D. Yau, Resonance of high-speed trains moving over a series of simple or
continuous beams with non-ballasted tracks, Engineering Structures 143 (2017) 295-305.
70. Q. Zeng, Y. B. Yang and E. G. Dimitrakopoulos, Dynamic response of high speed vehicles
and sustaining curved bridges under conditions of resonance, Engineering Structures 114
(2016) 61-74.
157
71. A. Littlefield, E. Kathe, R. Messier and K. Olsen, Gun Barrel Vibration Absorber to Increase
Accuracy, 19th AIAA Applied Aerodynamics Conference. AIAA-2001-1228: pp. 1228.
Seattle, WA, April 16-19, 2001.
72. E. Kathe, Lessons Learned on the Application of Vibration Absorbers for Enhanced Cannon
Stabilization, Shock and Vibration 8 No. 3-4 (2001) 131-139.
73. M. Tawfik, Dynamics and Stability of Stepped Gun-Barrels with Moving Bullets, Advances
in Acoustics and Vibration 2008 (2008).
74. L. Tumonis, S. Markus, K. Rimantas and K. Arnas, Structural Mechanics of Railguns in the
Case of Discrete Supports, IEEE Transaction on Magnetics 45 No. 1 (2009) 474-479.
75. K. Daneshjoo, M. Rahimzadeh, R. Ahmadi and M. Ghassemi, Dynamic Response and
Armature Critical Velocity Studies in an Electromagnetic Railgun, IEEE Transactions on
Magnetics 43 No. 1 (2007) 126-131.
76. I. Esen and A. Mehmet, Optimization of a Passive Vibration Absorber for a Barrel Using the
Genetic Algorithm, Expert Systems with Applications 42 No. 2 (2015) 894-905.
Abstract (if available)
Abstract
A semi-analytical method, based on distributed transfer function method (DTFM) and generalized assumed-mode method, is developed for modeling and dynamic analysis of coupled structure-moving subsystem problem. This newly developed method has a variety of engineering applications including elevated railways and highway ramps, cable transportation system, fast tube transportation and weaponry systems. Also devised in this effort is an analytical method of mapping transformation that can to predict the parametric resonance of a structure induced by repeatedly passing subsystems. ❧ The system in consideration is a coupled distributed-lumped system. The distributed system is a flexible supporting structure, which in this work is modeled as a stepped multi-span beam with flexible column supports. The lumped systems are moving subsystems, such as oscillators or rigid bodies, which pass over the supporting structure. The distributed and lumped systems are coupled at their interconnecting points by springs and dampers. The coupled system is governed by a mix set of one partial differential equation and several ordinary differential equations. Due to time varying locations of moving subsystems, the coupled system is nonautonomous or time-variant whose general analytical solution is not accessible. What’s more, flexible coupling between the structure and moving subsystems is extremely complicated because the number of contact points for one subsystem, and total number of subsystems in contact with the structure are generally time varying. To have a consistent and systematical formulation for structure-vehicle interaction, an extended solution domain (ESD) technique is developed. The ESD is a union of three domains, the beam domain and two virtual domains. The structure is extended to the virtual domains as rigid surfaces. Within the ESD, all moving subsystems are coupled with the “extended structure”, rendering a fixed number of DOFs and constant matrix dimensions in formulation. ❧ To solve the dynamic response of the coupled system, a semi-analytical method, that constituted of DTFM and generalized assumed-mode method, is developed. Analytical and closed form eigenfunctions (mode shapes) of the multi-span beam is obtained with augmented DTFM formulation. The coupled system is discretized via application of generalized assumed-mode method with analytical mode shapes being comparison functions. Because of this, the proposed method is proved to be much more efficient than conventional FEM. Numerical results reveal that vibration of the structure is dominated by static deflection caused by equivalent force if subsystems are moving relatively slowly. Dynamic interaction between the structure and moving subsystems will be significantly increased if subsystems are moving fast. Vibration of the structure with ever-increasing and significantly large amplitude can be induced by repeated passage of moving subsystems, which is identified as parametric resonance. ❧ Parametric resonance is different from conventional resonance where the frequency of external excitation matches with one of system’s natural frequencies. It is caused by repeated changing of system configuration, which is the repeated passage and coupling of moving subsystems for the problem studied in this work. A semi-analytical prediction method is developed based on mapping transformation. Quantitative resonance criterion is established by evaluating the spectral radius of a mapping matrix derived for the coupled system. Evaluating of the mapping matrix only requires simulation for one period of passage. This makes the proposed method extremely efficient in determination of parametric resonance. The proposed method is capable to predict not only the resonance condition, but also the steady state value for a bounded response. Numerical results show that the parametric resonance induced by a sequence of moving subsystems highly depends on system parameters, such as speed and spacing distance of subsystems, and cannot be accurately predicted by using a one-mode approximation for the beam structure.
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University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Gao, Hao
(author)
Core Title
Modeling and dynamic analysis of coupled structure-moving subsystem problem
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Mechanical Engineering
Publication Date
08/05/2020
Defense Date
04/30/2019
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
analytical resonance criterion,fast transporting subsystem,moving oscillator,moving rigid body,multi-span structure,OAI-PMH Harvest,parametric resonance,semi-analytical solution
Format
application/pdf
(imt)
Language
English
Contributor
Electronically uploaded by the author
(provenance)
Advisor
Yang, Bingen (
committee chair
), Flashner, Henryk (
committee member
), Wellford, Carter (
committee member
)
Creator Email
haogao@usc.edu,haogao111@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c89-206878
Unique identifier
UC11662801
Identifier
etd-GaoHao-7625.pdf (filename),usctheses-c89-206878 (legacy record id)
Legacy Identifier
etd-GaoHao-7625.pdf
Dmrecord
206878
Document Type
Dissertation
Format
application/pdf (imt)
Rights
Gao, Hao
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
analytical resonance criterion
fast transporting subsystem
moving oscillator
moving rigid body
multi-span structure
parametric resonance
semi-analytical solution